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LIBRARY 

OF  THE 

University  of  California. 


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OIF^T  OF 


v^ 


13 


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Class 


I 


Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

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http://www.archive.org/details/elementsofgeometOOrobirich 


ELEMENTS 


or 


GEOMETRY, 


PLANE  AND  SPHERICAL  TEIGONOMETEY, 


AND 


CONIC     SECTIONS 


BT 


H.    N.    ROBINSON,    A.    M., 

AUTHOR    OF    A    TREATISE    ON    ARITHMETIC,    AN    ELEMENTARY    AND     A      UNIVERSITT 
EDITION  OF  ALGEBRA,  A  WORK  ON   NATURAL   PHILOSOPHY,  AND  TWO   SEPARATE 
WORKS   ON    ASTRONOMY. 


jriFTEENTH     EDITION, 


NEW   YORK: 

IVISON    &    PHINNEY,    321    BROADWAY 

CINCINNATI:      JACOB      ERNST. 
CHICAGO :   S.  C.  GRIGGS  &  CO.,  89  <fc  41  LAKE  ST. 

ST.  LOUIS :  KEITH  X  WOODS.  BUFFALO:  FHIKNXT  te  00. 


1858. 


^      ^   OV  THE 

i   UNIVERSITY 


5^ 


Entered  according  to  act  of  Congress  in  the  year  1850,  by 

H.    N.    ROBINSON, 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States,  for  tho 

District  of  Ohio. 


•TKBEOTTTBO  BT  A.  C.  JAMM. 
CUTCnrilATI. 


PREFACE 


Ak  attempt  is  made  in  this  volume,  to  bring  the  science  of  geometry, 
directly  to  the  comprehension  of  the  learner ;  and  to  accomplish  thia 
end,  it  is  necessary  to  sweep  away  some  of  the  rubbish  and  some  of  the 
redundancies  which  have  seemed  only  to  obstruct  our  progress  and 
becloud  our  vision. 

All  attempts  to  prove  what  is  perfectly  obvious  to  every  one  without 
proof,  only  weakens  the  mind  rather  than  strengthens  it,  and  hence,  we 
have  discarded  all  such  propositions  as  the  following :  "  All  right  an- 
gles are  equal."  "  Any  two  sides  of  a  triangle  are  greater  than  the 
third  side."  "  Parallel  lines  can  never  meet,  however  far  they  may  be 
produced  " — and  some  few  others  of  like  character.  In  almost  every 
treatise  on  Geometry,  the  first,  or  one  of  the  first  propositions  for  de- 
monstration is,  "  That  all  right  angles  are  equal."  This  proposition  at 
once  excites  in  the  mind  of  the  intelligent  pupil,  a  mingled  sensation 
of  disappointment  and  indignation, — disappointment,  because  he  ex- 
pected to  learn  new  truths  ;  indignation,  because  he  feels  as  if  his 
time  and  common  sense  are  trifled  with. 

When  he  attempts  the  demonstration,  he  either  has,  or  has  not,  a 
correct  idea  of  a  right  angle  ;  if  he  has  a  correct  idea,  he  cannot  demon- 
strate, or  say  anything  that  can  be  called  a  demonstration — because 
the  proposition  is  all  embraced  in  the  definition  of  a  right  angle. 

If  he  has  not  the  correct  idea  of  the  term  right  angle,  he  must 
obtain  it  before  he  can  commence  any  demonstration;  so,  in  either 
case,  the  proposition  is  worse  than  useless. 

When  he  comes  to  the  proposition,  that  "  Any  two  sides  of  a  tri- 
angle, are  together,  greater  than  a  third  side,"  and  is  carried  through  a 
useless  demonstration,  he  looks  about  in  wonder  and  perplexity,  to 
discover  why  it  is  that  he  should  be  dragged  through  formal  techical- 
ities  to  arrive  at  the  perfectly  axiomatic  truth,  that  a  straight  line  is  the 
shortest  distance  between  two  points. 

11189;:' 


iv  PREFACE. 

Where  is  the  logic  of  proving  that  parallel  lines  will  never  meet, 
however  far  they  may  be  produced,  when  the  very  meaning  of  the  term 
parallel  is,  that  they  cannot  meet ;  hence,  we  say  that  all  attempts  to 
prove  what  is  perfectly  obvious,  tend  more  to  confuse  and  weaken, 
than  to  strengthen  and  enlighten. 

Notwithstanding  we  have  discarded  such  ^  like  propositions,  we  have 
omitted  none  of  the  truths  therein  expressed ;  for  we  have  put  them 
either  in  the  axioms  or  definitions,  and  have  made  as  complete  a  chain 
of  geometrical  truths  as  are  to  be  found  in  any  other  work. 

At  the  same  time,  no  attempt  has  been  made  to  present  all  the 
known  propositions  in  geometry  ;  we  have  taken  such  only  as,  united 
and  combined,  will  give  the  pupil  complete  power  over  the  science, 
and  make  hie  geometrical  knowledge  efficient,  useful,  and  practical. 

In  the  mathematical  sciences,  it  is  necessary  to  be  more  or  less 
technical,  formal,  and  exact ;  but  we  have  made  efforts  not  to  be  un- 
pleasantly so.  We  have  presumed  that  the  reader  will  exercise  his 
own  judgment  in  construing  our  language  ;  and  in  place  of  the  precise- 
ness  of  the  professor,  we  have  aimed  to  take  the  more  wholesome  and 
elevated  tone  of  the  practical  common-sense  man  of  the  world.  For 
the  sake  of  perspicuity  and  brevity,  we  have  freely  used  the  algebraic 
language  ;  and  the  whole  work  supposes  that  the  reader  clearly  compre- 
hends simple  equations,  and  is  able  to  perform  all  ordinary  operations 
with  them  ;  but  this  should  be  no  objection  to  the  use  of  this  book — for 
no  treatise  on  Geometry  should  be  studied  prior  to  Algebra,  whatever  be 
the  tone  and  style  of  the  Geometry. 

To  most  persons.  Geometry  is  a  very  dry  and  uninteresting  study;  and 
from  the  nature  of  the  human  mind  it  must  be  so,  until  the  pupil  catches 
the  spirit  of  the  science;  but  as  a  general  thing  that  spirit  cannot  be 
infused  until  some  essential  advancements  have  been  made;  hence, 
the  ill  success  of  many  who  undertake  this  study. 

It  is  essential  that  the  teacher  should  have  a  clear  view  of  all  these 
particulars  ;  that  he  should  possess  the  true  spirit  himself ;  and  then  he 
will  be  able  to  animate,  encourage,  and  assist  the  new  beginner,  until 
the  daylight  of  the  science  breaks  in  upon  his  mind. 

It  is  of  little  use  to  commence  Geometry  unless  the  learner  is  deter- 
mined to  go  through,  at  least,  so  far,  as  to  understand  Plane  Trigonom- 
etry. The  first  propositions  are  only  so  many  letters  in  the  great 
alphabet  of  science,  and  we  must  be  able  to  put  them  together,  before 
.we  can  really  perceive  their  utility  and  power.  These  considerations 
induced  us  to  be  very  full  and  practical  in  the  application  of  Geometry, 
and  if  a  student  can  go  through  this  book  understandingly,  we  are  sure 
that  his  geometrical  knowledge  will  be  at  once  ample  and  efficient. 


PREFACE.  T 

With  proper  encouragement  and  proper  instruction,  the  learner  will 
begin  to  discover  the  beauties  of  geometrical  demonstrations,  after 
passing  through  the  first  three  books,  and  when  that  discovery  is  made, 
all  serious  difficulties  will  be  over.  Yet  the  pupil  should  not  stop  there  ; 
for,  to  receive  the  benefits  of  any  science,  we  must  have  command  over 
that  science.  To  receive  the  benefits  of  any  enterprise,  we  must  carry 
it  through  to  completion,  or  be  content  to  lose  a  part,  if  not  the  whole 
of  our  labors  ;  it  is  emphatically  so  with  this  science. 

The  infinitesimal  system  has  been  used  in  demonstrations  to  a  greater 
extent  in  this,  than  in  most  other  works  of  like  kind,  and  although  the 
method  has  been  objected  to,  the  objections  are  neither  far-sighted  nor 
philosophical ;  a  rejection  of  this  method  necessarily  rejects  the  dif- 
ferential and  integral  calculus,  and  all  works  based  upon  them  as 
unscientific  and  unsound. 

In  plane  and  spherical  trigonometry,  great  pains  have  been  taken  to 
show  the  theoretical  beauties  of  those  sciences,  as  well  as  their  practi- 
cal application,  and  for  this  end,  many  of  the  demonstrations  have 
been  given  both  analytically  and  geometrically.  In  applying  these 
sciences,  more  examples  are  given  in  this  work  than  any  other  that 
I  have  seen,  and  such  questions  and  such  problems  have  been  chosen, 
as  to  show  the  great  power  and  utility  of  geometrical  science.  In 
confirmation  of  this,  we  refer  the  reader  to  the  various  astronomical 
problems,  and  in  particular  to  the  one,  giving  general  directions  for 
computing  the  beginning  or  end  of  a  local  solar  eclipse. 

Those  only  who  pay  particular  attention  to  Geometry,  will  be  able  to 
demonstrate  the  propositions  proposed  for  exercises  on  pages  100-104  ; 
they  are  designed  for  amateurs  in  particular  ;  they  are  marks  of  attain- 
ment to  which  all  may  aspire,  but  as  a  general  thing  they  will  require 
more  time  and  attention  than  can  be  devoted  to  them  in  schools  ;  there- 
fore, no  attempt  should  be  made  to  solve  all  of  them,  before  passing  on. 

In  conic  sections  we  have  not  been  as  full  as  some  other  treatises, 
especially  in  respect  to  the  hyperbola,  and  the  reason  for  our  brevity 
on  that  curve  is,  that  it  is  of  little  or  no  practical  utility  ;  it  is  merely 
a  curve  of  mathematical  curiosity.  The  ellipse  and  parabola  have  im- 
portant relations  to  astronomy,  and  projectile  motions,  and  we  have 
taken  particular  care  to  demonstrate  those  properties  essential  to  their 
application,  and  further  than  this  would  exceed  our  design ;  but  we  have 
given  this  amply  and  fully  ;  yet  this  treatise  is  not  designed  to  super- 
sede the  study  of  these  curves  again  in  Analytical  Geometry,  and  if 
the  student  understands  the  demonstrations  here  given,  he  will  be 
able  to  pursue  analysis  with  great  power  and  facility. 


CONTENTS. 


PLANE  GEOMETRY. 


BOOK    L 

Fkg*. 

General  PrincipIeSi . .     9 

Theorems  in  reference  to  right  lines  and  angles, 15 

BOOK    II. 

Proportion, 43 

The  definition  of  the  term  ratio, 43 

BOOK    III. 
Theorems  mostly  in  relation  to  the  circle  and  to  the  measure  of  angles,. . .  61 

BOOK    IV. 
Problems — ^geometrical  constructions, 78 

BOOK    V. 

The  measurement  of  polygons  and  circles, 90 

Exercises  in  geometrical  investigation, 100 

Problems  requiring  the  aid  of  algebra  for  their  solution, 103 

BOOK   VI. 
On  the  intersection  of  planes, 109 

BOOK    VII. 
Solid  Geometry, 118 


viii  CONTENTS. 

PLANE   TRIGONOMETRY. 

Elementary  principles, 136 

On  the  computation  of  sines  and  tangents, 144 

Application  of  Plane  Trigonom'^try, 156 

Explanation  of  the  tables, 157 

Oblique  angled  Trigonometry, 163 

Application  of  Trigonometry  to  the  measuring  of  bights  and  distances,. .  168 

SPHERICAL   TRIGONOMETRY. 

Elementary  principles, 176 

Napier's  circular  parts, 186 

Oblique  angled  Spherical  Trigonometry, 187 

Application — solution  of  right  angled  spherical  triangles, 196 

Solution  of  oblique  angled  spherical  triangles, 203 

Application  of  Spherical  Trigonometry  to  Astronomical  Problems, 207 

How  to  manage  a  local  Solar  Eclipse, 212 

Miscellaneous  Astronomical  Examples, 215 

Lunar  Observations, 216 

Appendix  to  Trigonometry, 217 


CONIC    SECTIONS. 

The  Ellipse, 227 

The  Parabola, 247 

The  Hyperbola, 263 


,"  \  8  B  A  Rp 

^    OF  THE 

UNIVERSITY 

OF  , 


GEOMETRY. 


DEFINITIONS. 

1.  GEOMETRY  is  the  science  that  estimates  and  compares  dis- 
tances, positions,  and  magnitudes. 

2.  A  Point  is  position,  not  magnitude,  and  on  paper  it  is  repre- 
sented by  a  visible  dot,  thus     . 

3.  A  Line  is  length,  only.     The  extremities  of  a  line  are  points. 

4.  A  Right  Line  has  the  same  direction  in  every  part. 

5.  A  Curved  Line  is  continually  changing  its  direction. 

6.  A  Broken  or  Crooked  Line  changes  its  direction  at  intervals. 

7.  An  Angle  is  the  difference  in  the  direction  of  two  lines. 

Two  lines  drawn  from  the  same  point,  and  in  the  same  direction,  are  one 
and  the  same  line. 

To  make  an  angle  apparent,  the  two  lines  must  C 

meet  in  a  point,  as  AB,  and  AC,  which  meet  at  the 
point  A, 


Two  lines,  not  having  the  same  direction,  and  not 
meeting  in  a  point  as  AB,  and  CD,  still  have  an 
angle  existing  between  them  equal  to  the  difference  in 
their  direction ;  and  to  make  the  angle  apparent, 
take  any  point  in  one  of  the  lines,  as  C,  and  con- 
ceive CH  to  lie  in  the  same  direction  as  AB.  Then 
the  difference  in  the  directions  of  CD  and  CH  mea- 
sures the  angle  ;  or  measures  the  difference  in  the 
directions  of  AB  and  CD, 


10 


GEOMETRY. 


8.  Angles  are  measured  by  the  number  of  degrees  of  a  circle 
included  between  the  two  lines  which  form 
the  angle  at  the  center  of  the  circle.  Thus, 
the  portion  of  the  circle  between  the  lines 
CA  and  CB  measures  the  angle  at  the 
center  of  the  circle.  Every  circle  is  di- 
vided into  360^,  and  the  greater  the  num- 
ber of  degrees  between  any  two  lines 
running  from  the  center,  the  greater  the 
angle. 

Angles  are  more  indefinitely  distinguished  by  AcviCt  Obtuse,  and 
riffkt  angles. 

9.  A  Hiffht  Angle  is  formed  by  one  line 
meeting  another  so  as  to  make  equal  angles 
with  the  other  line. 

One  line  so  inclined  to  another  is  said  to 
be  perpendicular  to  another. 

10.  An  Acute  Angle  is  less  than  a  right 
angle. 


1 1 .  An  Obtuse  Angle  is  greater  than  a 
right  angle. 


12.  An  angle  is  named  by  a  letter  at  its  vertex, 
as  A.  When  two  or  more  angles  have  their  ver- 
tices at  the  same  point,  this  method  will  not  be 
sufficiently  definite. 

Thus,  when  several  lines  as  AB,  AC,  AD, 
all  meet  at  the  point  A,  several  angles  are 
formed ;  and  to  define  the  one  formed  by  the  two 
lines  AB  and  A  C,  we  must  say  the  angle  CAB, 
or  BA  C.  To  express  the  angle  requires  three 
letters,  and  the  middle  one  must  be  at  the  vertex 
of  the  angle.  The  angle  DAC  is  the  angle  made  by  the  two 
lines  DA  and  A  C,  The  angle  DAB  is  the  angle  made  by  the 
two  lines  DA  and  AB* 


DEFINITIONS.  n 

13.  Two  lines  similarly  situated  and  making  equal  angles  with  a 
third  line,  all  being  in  the  same  plane,  are  paralleU 

Parallel  lines  may  be  either  right  lines,  as  ^  J?,  or  curved      ^  ^ 

lines,  as  C  D  ;  but  at  present  we  are  only  considering  right 
lines. 

Rectilinear  parallels  have  the  same  absolute  direction ; 
and,  conversely,  lines  having  the  same  absolute  direction,  are  parallel. 

Two  parallel  lines  cannot  be  drawn  from  the  same  point ;  for  io  fulfill  the  con- 
dition of  parallelism,  any  attempt  to  draw  them  would  run  them  into  the  same 
direction,  and  thus  make  one  line.  Conversely,  then,  two  parallel  lines  cannot 
meet  in  a  point,  however  far  they  may  be  produced. 

14.  Superficies  are  either  Plane  or  Curved. 

A  Plane  Superficies,  or  a  Plane,  is  that  with  which  a  right  line 
may  every  way  coincide.  Or,  if  the  line  touch  the  plane  in  two 
points,  it  will  touch  it  in  every  point ;  but,  if  not,  it  is  curved. 

15.  Plane  figures  are  bounded  either  by  right  lines  or  curves. 

16.  Plane  figures  that  are  bounded  by  right  lines  have  names 
according  to  the  number  of  their  sides,  or  of  their  angles ;  for 
they  have  as  many  sides  as  angles  ;  the  least  number  being  three. 

17.  A  figure  of  three  sides  and  angles  is  called  a  triangle  ;  and 
it  receives  particular  denominations  from  the  relations  of  its  sides 
and  angles. 

18.  An   Equilateral    Triangle   has  three  equal 
sides. 

19.  An  Equiangular  Triangle   has  three   equal 
angles. 

Every  Equilateral  Triangle  is  also  Equiangular. 

20.  An  Isosceles  Triangle  has  two  equal  sides. 

21.  A  Right  Angled  Triangle  has  one  right  angle. 

22.  An  Obtuse  Angled  Triangle  has  one  obtuse  angle. 

23.  An  Acute  Angled  Triangle  has  all  its  three  angles  acute. 

24.  A  Quadrilateral  figure  has  four  sides  and  four  angles. 

25.  A  Parallelogram  is  a  quadrilateral  which  has  its  opposite 
sides  parallel,  and  it  may  take  the  name  of  rectangle,  square,  rhom- 
ioid,  or  rhombus,  according  to  the  relation  of  its  sides  and  angles. 

26.  A  Rectangle  is  a  parallelogram,  having 
its  angles  right  angles. 


12 


GEOMETRY 


27.  A  Square  has  all  its  sides  equal,  and  all  its 
angles  right  angles. 


28.  A  Rhomboid  is  an  oblique  angled 
parallelogram. 


29.  A  Rhombus  is  an  equilateral  rhomboid 


30.  A  Trapezium  is  any  irregular  quadrilateral. 


31.  A  Trapezoid  is  a  quadrilateral  which  has  two  opposite 
sides  parallel. 

32.  A  figure  of  five  sides  is  called  a  Pentagon  ;  of  six,  a 
Hexagon ;  of  eight,  an  Octagon,  &c. ;  but  all  these  figures  are  in 
general  called  Polygons. 

33.  Diagonals  are  lines  joining  any  two  angles  of  a  polygon  not 
adjacent. 

34.  Polygons  may  be  similar  without  being 
equal ;  that  is,  the  angles  and  the  number  of 
sides  equal,  and  the  length  of  the  sides  and 
the  size  of  the  figures  unequal. 

35.  A  Perimeter  of  any  figure  is  the  sum  of  all  its  sides. 

36.  The  Altitude  of  any  figure  is  ihe  perpendicular  distance  from 
any  side,  or  any  angle,  to  the  opposite  side  or  angle. 

37.  A  Circle  is  a  figure  bounded  by  one 
uniform  curved  line,  and  a  certain  point 
within  it,  from  which  all  straight  lines 
drawn  to  the  curve  are  equal,  and  this 
point  is  called  the  center. 


DEFINITIONS.  13 

EXPLANATION   OF  TERMS. 

1.  A  Postulate  is  a  position  taken ;  a  fact  that  must  be  admitted. 

2.  An  Axiom   is  a  self-evident  truth ;   not  only  too  simple  to 
require,  hui  too  simple  to  admit,  of  demonstration, 

3.  A  Proposition  is  something  which  is  either  proposed  to  be 
done,  or  to  be  demonstrated,  and  is  either  a  problem  or  a  theorem. 

4.  A  Problem  is  something  proposed  to  be  done. 

5.  A  Theorem  is  something  proposed  to  be  demonstrated. 

6.  A  Lemma  is  something  which  is  premised,  or  demonstrated, 
in  order  to  render  what  follows  more  easy. 

7.  A  Corollary  is  a  consequent  truth  gained  immediately  from 
some  preceding  truth  or  demonstration. 

8.  A  Scholium  is  a  remark  or  observation  made  upon  something 
going  before  it. 

POSTULATES. 

1.  Let  it  be  granted  that  a  straight  line  can  be  drawn  from  any 
one  point  to  any  other  point. 

2.  That  a  straight  line  can  be  produced  to  any  distance,  or  ter- 
minated at  any  point. 

3.  That  a  circle  can  be  drawn  from  any  center,  at  any  dis- 
tance from  that  center. 

AXIOMS. 

1 .  Things  which  are  equal  to  the  same  thing  are  equal  to  each  other, 

2.  Wlien  equals  are  added  to  equals  the  wholes  are  equal. 

3.  When  equals  are  taken  from  equals  the  remainders  are  equal. 

4.  When  equals  are  added  to  unequals  the  wholes  are  unequal. 

5.  When  equals  are  taken  from  unequals  the  remainders  are  unequal. 

6.  Things  which  are  double  of  the  same  thing,  or  equal  things,  are 
fiqual  to  each  other. 

7.  Things  which  are  halves  of  the  same  thing  are  equal. 

8.  Every  whole  is  equal  to  all  its  parts  taken  together. 

9.  Things  which  coincide,  or  fill  the  same  space,  are  identical,  or 
mutually  equal  in  all  their  parts. 

10.  All  right  angles  are  equal  to  one  another. 

1 1 .  Two  straight  lines  cannot  inclose  a  space. 

12.  A  straight  line  is  the  shortest  distance  between  two  points. 

13.  The  wJiole  is  greater  than  its  'part. 


14 


GEOMETRY. 
ABBREVIATIONS. 


The  common  algebraical  signs  will  be  used  in  this  work,  and 
demonstrations  will  sometimes  be  made  through  the  medium  of 
equations ;  and  it  is  so  necessary  that  the  student  in  Geometry 
should  understand  some  of  the  more  simple  operations  of  Algebra, 
that  we  suppose  he  is  acquainted  with  the  use  of  the  signs.  As 
the  words  circle,  angle,  triangle,  hypothesis,  axiom,  are  constantly 
occurring  in  a  course  of  Geometry,  we  shall  abbreviate  them  as 
follows : 


Addition  is  expressed  by  . 
Subtraction       "         ««  .         . 

Multiphcation  '*         "      . 
Equality  "         «* 

Cheater  than     **         **      •         • 
Less  than  <«         «<  ,         , 

Thus  :  B  is  greater  than  A,  is  written 

B  is  less  than  A,       "       " 
Let  a  circle  be  expressed  by     . 
An  angle  by        **  " 

A  triangle  by       "  •*       .         . 

The  word  hypothesis         ** 
Axiom  is  expressed  " 

Theorem  "  «* 

Corollary         **  "        .        . 

Perpendicular "  ** 

When  the  difference  of  two  quantities  is  e 

out  knowing  which  is  the  greater, 

lowing  symbol,      .        • 


.    ByA. 

B<^A, 

.     o. 

J. 

.     A. 

.         .      (hy.) 

(ax.) 

.         .      (th.) 

(Cor.) 

J.. 

xpressed,  with- 

we  use  the  fol- 


BOOK    I.  U 


BOOK      I. 

THEOREM     1. 

When  one  line  meets  another^  the  turn  of  the  two  angles  which  it 
makes  on  the  same  side  of  the  other  line,  is  equal  to  two  right  angles. 

Let  AB  meet  CD;   then  we  are  to  de- 
monstrate that  the  two  angles  ABD4-ABC=  j  ^ 
two  right  angles,                                                                   { 

If  AB  does  not  incline  on  either  side 

of   CD  and  the  angle  ABD=ABC,  then      

these  angles  are  right  angles  by  definition  9.      ^ 

But  if  these  angles  are  unequal,  conceive  the  dotted  line,  BE, 
drawn  from  the  point  B,  so  as  not  to  incline  on  either  side  ;  then 
by  the  definition,  the  angles  CBU  and  UBD  are  right  angles ; 
but  the  angles  CBA-{-ABD  make  the  same  sum,  or  fill  the 
same  angular  space,  as  the  two  angles  CBE  and  EBD ;  there- 
fore, CBA-^-ABD^Ufo  right  angles.     Q.  E.  D.  * 

Cor.  1 .  Hence,  all  the  angles  which  can  be  made  at  any  point 
J5,  by  any  number  of  lines  on  the  same  side  of  the  right  line  CD, 
are,  when  taken  all  together,  equal  to  two  right  angles. 

Cor.  2.  And,  as  all  the  angles  that  can  be  made  on  the  other 
side  of  the  line  CD  are  also  equal  to  two  right  angles,  therefore 
all  the  angles  that  can  be  made  quite  round  a  point  B,  by  any 
number  of  Imes,  are  equal  to  four  right  angles. 

Cor.  3.  Hence,  also,  the  whole  circumference 
of  a  circle,  being  the  sum  of  the  measures  of  all 
the  angles  that  can  be  made  about  the  center 
F,  (def.  8),  is  the  measure  of  four  right  angles ; 
consequently,  a  semicircle,  or  180  degrees,  is 
the  measure  of  two  right  angles ;  and  a  quadrant,  or  90  degrees, 
the  measure  of  one  right  angle. 


Tbe  initials  of  a  Latin  phrase,  meaning  "  which  wa$  to  be  demoruiraied." 


m 


GEOMETRY. 


THEOREM    2. 

Jjf  one  straight  line  meets  two  other  straight  lines  at  a  common  pointy 
forming  two  angles,  which  together  make  two  right  angles,  the  two 
straight  lines  are  one  and  the  same  line. 

If  AB  meets  the  two  lines  DB 
and  BO  at  the  common  point  B, 
and  the  two  angles  DBA-^-ABO 
=two  right  angles,  then  we  are  to 
demonstrate  thai  DB  and  BC  form 
one  and  the  same  straight  line. 

If  DB  and  BO  are  not  in  the 
same  line,  produce  DB  to  jE>  making  a  continued  line  DH :  then 
by  (th.  1)  the  angles 

ABD-\-ABU=  2B     (2  R  indicates  two 

But  by  (hy.)  ABD-{-AB0=2R     right  angles.) 

By  subtraction        ABU— AB  0=0 

That  is,  the  angle  OBJS  is  zero  ;  and  DBO  is  a  continued  line  ; 
or  BO  falls  on  BK  Q.  JE.  D, 

THEOREM    3. 

J^  two  straight  lines  intersect  each  other,  the  opposite  vertical  angles 
are  equal. 

If  AB  and  CD  intersect  each  other 
at  SI,  we  are  to  demonstrate  that  the  angle 
AEG  equals  its  opposite  angle  DEB,  and 
AED  =  CEB. 

As  ABB  is  a  right  line,  ^A  is  ex- 
actly in  the  opposite  direction  from  UB ;  and  for  the  same  reason 
S!0  is  opposite  in  direction  from  UD ;  therefore,  the  difference  in 
direction  between  EA  and  jE'C  is  equal  to  the  difference  in  direction 
between  UB  and  UD;  or  by  (def.  7),  the  angle  AEC^DEB.  In 
the  same  manner  we  can  show  that  the  angle  AED=  OEB.   Q.  E.  D. 

Otherwise  :  Let  AEO=z,  AED=y,  and  DEB=x;  then  we  are 
to  show  that  x=z.  As  AB  is  a  right  line,  and  DE  falls  upon  it, 
we  have,  by  (th.  1),  x-\-y=2li 

Also,        .         .        .         .z-\-y=2B 

By  subtraction. 
By  transposition. 


.x—z=0 

.       x=z     Q.E.D, 


BOOK    I.  17 

THEOREM    4. 

If  a  atraigJU  line  falls  across  two  parallel  straight  lines  y  the  sum  of 
the  two  interior  angles  on  the  same  side  of  the  crossing  line  is  eqital  to 
two  right  angles. 

Let  AB  and  CD  be  two  paral- 
lel lines,  and  EF  running  across 
them ;  then  we  are  to  demonstrate 
that  the  angle  BGH+GHD=2R. 
Because  Q-B  and  HD  are  parallel, 
they  are  equally  inclined  to  the  line 
EFy  or  have  the  same  difference  of 
direction  from  that  line  :  Therefore  J  FGB=  J  GffD.  To  each 
of  these  equals  add  the  J  BGH. 

Then  FGB-^BGH=  GHD-\-BGH. 

But  by  (th.  1 )  the  first  member  of  this  equation  is  equal  to  two 
right  angles  :  that  is,  the  two  interior  angles  GHD  and  BGB  are 
together  equal  to  two  right  angles.      §.  E,  JD, 

THE  OREM   5. 

Jf  a  straight  line  falls  across  two  parallel  straight  lines,  the  interior 
alternate  angles  are  equal;  and  also  the  opposite  exterior  angles. 

On  the  supposition  that  AB  and   CD  are  parallel^  (see   last 
figure),  and  EF  falls  across  them,  we  are  to  demonstrate 
1st.  That  the  J  ^6^^=the  alternate  J  GffD. 
2d.   That  A  GF=^EHD  ;  or  FGB^  CHE. 
By  the  definition  of  parallel  lines  we  have 
FGB^GHD 
But  FGB=AGff  (ih.  3) 
Hence  AGH=GHD  (ax.  1)     Q.  E.  D. 
2d.  The  J   FGB=GHD.     But  GHD=CHE  (th.  3);  there- 
fore, FGB=^  CHE.     In  the  same  manner  we  prove  that  A  GF  is 
equal  to  EHD.     Q.  E.  D, 

THEOREM    6. 

If  a  straight  line  falls  across  two  parallel  siraiglU  lines,  the  exterior 
angles  are  equal  to  the  interior  opposite  angles  on  the  same  side  of  the 
crossing  line. 
2 


IS  GEOMETRY. 

If  AB  and  CD  are  parallel,  (see  last  figure),  and  UF  crosses 
them,  then  we  are  to  prove  that  the  exterior  j  FGB=  QHD 
And  .        .        .        .      AGF=CHa 
For    ...        .       AGH=FGB  (th.  3) 
Also.        .        .        .       AGff=GJII)  {th.  5) 
Hence    FGB=GIID  (ax.  I) 
In  the  same  manner  we  prove  that  A  GF=  CHG.     Q.  E.  D. 

THEOREM    7. 

If  a  straight  line  falls  across  two  other  straight  lines,  and  makes  the 
mm  of  the  two  interior  angles  on  the  same  side  equal  to  two  right 
angles,  the  two  straight  lines  must  be  parallel. 

Let  ^i^  be  the  line  falling  across 
the  lines  AB  and  CD,  making  the 
two  angles  BGE-{-GBD=^totwo 
right  angles  ;  then  we  are  to  demon- 
strate that  AB  and  CD  must  be 
parallel. 

As  FF  is  a  right  line,  and  BA 
meets  it,  the  two  angles  (th.  1 ) 

FGB+BGII=2li 

By  (hy.)     .  GHD^BGH=^R 

By  subtraction,  FGB—GHD—^,  That  is,  there  is  no  differ- 
ence in  the  direction  of  GB  and  HD  from  the  same  line  EF;  but 
when  there  is  no  difference  in  the  direction  of  lines  (def  13)  the 
lines  are  parallel ;  therefore,  AB  and  CD  are  parallel.    Q.  E.  D. 

THEOREM    8. 

Parallel  lines  can  never  meet,  however  far  they  may  be  produced. 

If  the  lines  AB  and  CD  (see  last  figure)  should  meet  at  any 
distance  on  either  side  of  EF,  they  would  there  form  an  angle ; 
and  if  they  formed  an  angle  they  would  not  run  in  the  same  direc- 
tion ;  and  not  running  in  the  same  direction,  they  would  not  be 
parallel ;  but  by  (hy.)  they  are  parallel ;  therefore  they  cannot 
meet     §.  E.  D, 


BOOK    I 


19 


THEOREM    9 


If  two  straigU  lines  are  parallel  to  a  third,  they  are  parallel  to 
each  other. 

If  AJB  is  parallel  to  UF,  and 
CD  also  parallel  to  JEFy  then  we 
are  to  show  that  AB  is  parallel  to 
CD. 

Because  AB  and  FFare  parallel, 
they  make  equal  angles  with  the 
line  HG  (def.  13,  2)  ;  and  because 
CD  and  FF  are  parallel,  those  two  lines  make  equal  angles  with 
the  line  II&. 

Hence  AB  and  CD,  making  equal  angles  with  another  line  that 
falls  across  them,  they  are  therefore  parallel  (def.  7).     Q.  F.  D. 


THEOREM     10 


If  two  angles  have  their  sides  parallel,  the  two  angles  will  he  equal. 

Let  the  two  angles  be  A  and 
DBF)  AC  parallel  to  DB,  and 
^^  parallel  to  BF. 

On  that  hypothesis  we  are  to 
prove  that  the  angle  A=DBF. 

Produce  DB,  if  necessary,  to 
meet  ^^in  O, 

Then        .         ^  DBF=J  DOff 

Also     .        .  JiA^ADQH 

Therefore  DBF=A 


(th.  6) 
(th.  6) 
(ax.  1) 


Q,  E.  D. 


Scholium,  When  ^ZT  extends  in  the  opposite  direction,  it  is  still 
parallel  to  BF ;  but  the  angle  then  is  the  supplemental  angle  to 
DBF)  that  is,  equal  to  FBQ, 


JO  GEOMETRY. 

THEOREM     11. 

If  any  side  of  a  triangle  be  produced,  the  exterior  angle  is  equal  to 
the  sum  of  the  two  interior  opposite  angles  ;  and  the  sum  of  the  three 
angles  is  equal  to  two  right  angles. 

Let  ABCh&  any  triangle.  Pro- 
duce AB  to  D.  Then  we  are  to 
show  that  the  angle  CBD=  _i  A 
-j-the  angle  C ;  also,  that  the  Bu- 
gles A-{- C-\- CBA=2li. 

From  B  conceive  BU  drawn 
parallel  U)  AC ; 

Then  EBD=  J.  A     (th.  6) 

By(th.  6)  CBE=j^  0    (alternate  angles). 

•    By  addition    j   CBD==A-\-C     Q.  E.  D. 

To  each  of  these  equals  add  the  angle  CBA,  and  we  have 

CBD-{-  CBA=A-{-  C7+  CBA 
But    .        .     CBD-\-CBA=2E  (th.  1) 

Therefore  A-{-C-{-  CBA=2B       (ax.  1 ) 

That  is,  the  three  angles  of  the  triangle  are,  together,  equal  to 
two  right  angles ;  and  this  triangle  represents  any  triangle  ;  there- 
fore, the  sum  of  the  three  angles  of  any  triangle  is  equal  to  two 
right  angles.     Q.  E.  B. 

Cor.  1 .  As  the  exterior  angle  of  any  triangle  is  equal  to  the  sum 
of  the  two  interior  and  opposite  angles,  therefore  it  is  greater  than 
either  one  of  them. 

Cor.  2.  If  two  angles  in  one  triangle  be  equal  to  two  angles  in 
another  triangle,  the  third  angles  will  also  be  equal,  (ax.  3),  and 
the  two  triangles  equiangular. 

Cor.  3.  If  one  angle  in  one  triangle  be  equal  to  one  angle  in 
another,  the  sums  of  the  remaining  angles  will  also  be  equal  (ax.  3). 

Cor.  4.  If  one  angle  of  a  triangle  be  right,  the  sum  of  the  other 
two  will  also  be  equal  to  a  right  angle,  and  each  of  them  singly 
will  be  acute,  or  less  than  a  right  angle. 

Cor.  5.  The  two  least  angles  of  every  triangle  are  acute,  or  each 
less  than  a  right  angle. 


BOOK    I. 


21 


THEOREM     12. 

In  any  quadrangle  the  sum  of  all  the  four  inward  angles  is  equal 
-to  four  right  angles. 

Let  ABCD  be  a  quadrangle  ;  then  the  sum 
of  the  four  inward  angles  A-\-B-\-  C-\-D  is  equal 
to  four  right  angles. 

Let  the  diagonal  AQ  he  drawn,  dividing  the 
quadrangle  into  two  triangles,  ABO,  ADC; 
then,  because  the  sum  of  the  three  angles  of  each  of  these  tri- 
angles is  equal  to  two  right  angles  (th.  11),  it  follows  that  the 
sum  of  all  the  angles  of  both  triangles  which  make  up  the  four 
angles  of  the  quadrangle,  must  be  equal  to  four  right  angles  (ax.  2). 
§.  E.  D. 

Cor.  1 .  Hence  if  three  of  the  angles  be  right  angles,  the  fourth 
will  also  be  a  right  angle. 

Cor.  2.  And  if  the  sum  of  two  of  the  four  angles  be  equal  to 
two  right  angles,  the  sum  of  the  remaining  two  will  also  be  equal 
to  two  right  angles. 

SCHOLIUM. 


^V-7^ 


In  any  figure  hounded  by  right  lines  and  angles,  the  sum  of  all  the 
interior  angles  is  equal  to  tioice  as  many  right  angles  as  the  figure  has 
sides,  less  four  right  angles. 

Let  ABCDE  be  any  figure;  then 
the  sura  of  all  its  inward  angles,  A-\- 
B-\-C-\-D-\-E,  is  equal  to  twice  as 
many  right  angles,  wanting  four,  as 
the  ficnire  has  sides. 

o 

For,  from  any  point  P,  within  it, 
draw  lines  PA,  PB,  PC,  <fec.,  to  all  the  angles,  dividing  the  poly- 
gon into  as  many  triangles  as  it  has  sides.  Now  the  sum  of  the 
three  angles  of  each  of  these  triangles,  is  equal  to  two  right  angles 
(th.  11) ;  therefore  the  sum  of  the  angles  of  all  the  triangles  is 
equal  to  twice  as  many  right  angles  as  the  figure  has  sides.  But 
the  sum  of  these  angles  contains  the  sum  of  four  right  angles  aboul 


23 


GEOMETRY. 


the  point  P :  take  these  away,  and  the  sum  of  the  interior  angles 
of  the  figure  is  equal  to  twice  as  many  right  angles  as  the  figure 
has  sides  less  four  right  angles.     Q.  E.  D. 

From  this  principle  we  can  deduce  the  following  rule  to  find  the 
sum  of  the  interior  angles  of  any  right-lined  figure  : 

Rule.  Subtract   2  from  the  number  of  sides,  and  multiply  the 
remainder  by  2,  and  the  product  will  be  the  number  of  right  angles. 

Thus,  if  the  sides  be  represented  by  5,  then  the  rule  gives 
(25 — 4)  ;  nor  is  the  rule  varied  in  case  of  a  re- 
entrant angle,  as  represented  at  d  in  the  figure  abed 
e  f.  Draw  the  dotted  hnes  from  the  angle  d  to  the 
several  opposite  angles,  making  as  many  triangles 
as  the  figure  has  sides,  less  two,  and  each  triangle 
has  two  right  angles  :  hence  the  rule. 


THEOREM     13. 

Two  triangles  which  have  two  sides,  and  the  included  angle  in  the 
one,  equal  to  the  two  sides  and  included  angle  in  the  other,  are  identical, 
or  equal  in  all  respects. 

In  two  As,  ABC  and  DEF,  on 
the  supposition  that  AB=DE,  and 
AC=DF,  and  the  _j  ^=  J  D,  we 
are  to  prove  that  BC  mws/=EF,  the 
J  B=  JE,  and  the  J  C=  J  F. 

Conceive  the  A  ABC cMi  out  of  the 
the  paper,  taken  up,  and  placed  on 
the  A  DEF  in  such  a  manner  that  the  point  A  shall  fall  on 
the  point  D,  and  the  line  AB  on  the  line  DE ;  then  the  point 
B  will  fall  on  the  point  E,  because  the  lines  are  equal.  Kow, 
as  the  J  A==  _\  D,  the  line  AC  must  take  the  same  direction  as 
DF,  and  fall  on  DF;  and  as  the  line  AC=DF,  the  point  (7  will 
fall  on  F.  B  being  on  E  and  (7  on  F,  BC  must  be  exactly  on  EF, 
(otherwise,  two  straight  hnes  would  enclose  a  space  ax.  11),  and 
BC=EF,  and  the  two  magnitudes  exactly  fill  the  same  space  ; 
therefore,  the  two  As  are  identical,  (ax.  9),  and  the  angle  B=E, 
and  C=F.     Q.  E.  D. 


BOOK    I  83 

THEOREM     14. 

When  two  triangles  have  a  side  and  two  adjacent  angles  in  the  one, 
equal  to  a  side  and  two  adjacent  angles  in  the  other,  the  two  triangles 
are  equal  in  all  respects. 

In  two  As,  as  ABO  and  Di:F, 
on  the  supposition  that  BC=EF, 
the  angle  B=E,  and  C=^F,  we  are 
to  prove  thai  AB=DE,  AC=DF,  and 
the  angle  A=D. 

Conceive  the  A  ABC  taken  up 
and  placed  on  the  A  BEF  so  that 
the  side  BC  shall  exactly  coincide  with  its  equal  side  EF; 
then  because  the  angle  B  is  equal  to  the  angle  E,  the  line  BA  will 
take  the  direction  of  ED,  and  fall  exactly  upon  it ;  and  because 
the  angle  C  is  equal  to  the  angle  F,  the  line  CA  will  take  the 
direction  of  FD,  and  exactly  fall  upon  it ;  and  the  two  lines  BA 
and  CA  e3»actly  coinciding  with  the  two  lines  ED  and  FD,  the 
point  A  will  fall  on  D,  and  the  two  magnitudes  exactly  fill  the 
same  space  ;  therefore,  by  (ax.  9)  they  are  identical,  and  AB=' 
ED,  AC=DF,  and  the  J  A=jD.     Q.  E,  D. 

THEOREM     15. 

Jftwo  sides  of  a  triangle  are  equal,  the  angles  opposite  to  these  side* 
will  be  equal. 

Let  AB  C  be  the  triangle  ;  and  on  the  suppo- 
sition that  A  C=  CB,  we  are  to  prove  that  the 
angle  A=B. 

Conceive  the  angle  C  divided  into  two  equal 
angles  by  the  line  CD;  then  we  have  two  As, 
ADC  and  CBD,  which  have  the  two  sides,  -4 C 
and  CD  of  the  one,  equal  to  the  two  sides,  CB 
and  CD  of  the  other ;  and  the  included  angle  A  CD,  of  the  one, 
equal  to  BCD  of  the  other:  therefore  (th.  13),  AD=BD,  and 
the  angle  A,  opposite  to  CD  of  the  one  triangle,  is  equal  to  the 
angle  B,  opposite  to  CD  of  the  other  triangle  :  that  is,  j  A 
«  J  JS.     Q.  E.  D. 


u 


GEOMETRY. 


Bk- -V^ 


Cor.  1.  As  the  two  triangles  A  CD  and  BCD  are  in  all  respects 
equal,  the  line  which  bisects  the  vertical  angle  of  an  isosceles  A 
also  bisects  the  base,  and  falls  perpendicular  on  the  base. 

Scholium.  Any  other  point  as  well  as  C  may  be  taken  in  the 
perpendicular  DCy  and  lines  drawn  to  the  extremities  A  and  £; 
such  lines  will  be  equal,  as  we  can  prove  by  theorem  15  ;  hence, 
we  may  announce  this  truth :  That  if  a  perpendicular  be  draum 
from  the  middle  of  a  line,  any  point  in' the  perpendicular  is  at  eqtcal 
distance  from  the  tux)  extremities. 

THEOREM    16. 

The  greater  side  of  every  triangle  has  the  greater  angle  opposite  to  iU 

Let  ABC  be  the  A;  and  on  the  supposition 
that  ACh  greater  than  AB,  we  are  to  prove  that 
the  angle  ABC  h  greater  than  the  _J  C.  From 
the  greater  of  the  two  sides  A  C,  take  AD,  equal 
to  ^-6  the  less,  and  join  BD;  thus  making  two 
triangles  of  the  original  triangle.  As  AB=ADf 
the  J  ADB^  the  J  ABD  (th.  15). 

But  the  J  ADB  is  the  exterior  angle  of  the  A  BDC,  and  there- 
fore greater  than  C :  that  is,  the  J  ABD  is  greater  than  the  angle 
C.     Much  more,  then,  is  the  angle  ^^(7  greater  than  C.    Q.  E.  D, 

THEOREM    17. 
If  two  triangles  have  two  sides  of  the  one  equal  to  two  sides  of  the 
other,  each  to  each,  and  an  angle  opposite  one  of  the  equal  sides  in  each 
triangle  equal,  then  mil  the  two  triangles  be  equal. 

Let  ABChQ  one  triangle  and  ADCi\\Q  other  in  which  AD=^ABy 
BC=DC,  and  the  angles  opposite  BC  and  i>C  equal,  then  will  the 
angle  ABC=ADC,  and  KG  be  a  converse  side. 

Place  the  two  A's  so  that  the  given  angles 
will  come  together  at  A,  and  lie  on  the  oppo- 
site sides  of  the  line  A  C. 

Then  because  AB=AD,  ABD  is  an  isos- 
celes A,  and  the  line  -4  (7  which  bisects  the 
angle  A  is  perpendicular  to  BD  and  bisects 
BD  (th.  1 5,  cor.  1 ).  Now  BCarndDC  must 
termuiate  in  the  same  point  C,  because  BC= 
DC  (th.  15,  scholium),  therefore,  AC  is 
common  to  the  two  A's  ABC,  ADC;  and 
the  A's  are  identical.  Q.  £J.  D. 
Scholium.  There  are,  in  fact,  two  cases  in  this  theorem,  because 
BC=BE,  and  DC=^DE,  giving  two  pair  of  A's. 


X  X 


Bkf [   -^D 


BOOK    I. 


35 


THEOREM    18. 

Th^  difference  of  any  two  sides  of  a  triangle  is  less  than  the  third 
side. 

Let  ABC  be  the  A,  and  let  ^C7  be  greater 
than  AB;  then  we  are  to  prove  that  A  G — AB 
is  less  than  BO. 

As  a  straight  line  is  the  shortest  distance  be- 
tween two  points, 

Therefore,     .        AB+B Cy  AC. 

From  these  unequals  subtract  the  equals 
AB=AB,  and  we  have  J5(7>  AC—AB.    (ax.  5).     Q.  E.  D. 


THEOREM      19. 

When  two  triangles  have  all  three  of  the  sides  in  one  triangle  equal 
to  all  three  in  the  other,  each  to  each,  the  two  triangles  will  be  identical 
and  have  equal  angles  opposite  equal  sides. 

In  two  triangles,  as  ABC  and 
ABDy  on  the  supposition  that  the 
side  AB  of  the  one =^.6  of  the 
other,  AC==AD,  and  BC=^BDy 
we  are  to  demonstrate  that  the  angle 
ACB=ihe  angle  ADB,  BAC= 
BAD,  and  ABC=ABD. 

Conceive  the  two  triangles  to  be  joined  together  by  their  longest 
equal  sides,  and  draw  the  line  CD. 

Then,  in  the  triangle  A  CD,  because  the  side  AC  is  equal  to 
AD  by  (hy.),  the  angle  A  CD  is  equal  to  the  angle  ADC  (th.  15). 
In  like  manner,  in  the  triangle  BCD,  the  angle  BCD  is  equal  to 
the  angle  BDC,  because  the  side  BC  is  equal  to  BD.  Hence, 
then,  the  angle  A  CD  being  equal  to  the  angle  ADC,  and  the 
angle  BCD  to  the  angle  BDC,  by  equal  additions  the  sum  of 
the  two  angles  A  CD,  B  CD,  is  equal  to  the  sum  of  the  two  AD  C, 
BDC (^x.  2)  ;  that  is,  the  whole  angle  ACB  is  equal  to  the  whole 
angle  BDA . 


26  GEOMETRY. 

Since  then  the  two  sides,  A  C,  CB,  are  equal  to  the  two  sides 
AD,  DBy  each  to  each,  by  (hy.),  and  their  contained  angles -4  C5, 
ADB,  also  equal,  the  two  triangles  ABQ,  ABB,  are  identical 
(th.  13),  and  have  their  other  angles  equal,  the  angle  BAC  to  the 
angle  BAB,  and  the  angle  ABC  to  the  angle  ABB.     Q.  E.  D, 

THEOREM    A. 

J^  there  he  two  triangles  which  have  the  two  sides  of  the  one  equal  to 
the  two  sides  of  the  other,  each  to  each,  and  the  included  angles  uneqiial, 
the  third  sides  will  he  unequal,  and  the  greater  side  unll  helong  to  the 
triangle  which  has  the  greater  included  angle. 

Let  ABC  be  one  A,  and  A  CD 
the  other  A-  Let  AB  and  AC  oi 
the  one  A  be  equal  to  AD  and 
AC  oi  the  other  A-  But  the 
angle  BA  C  greater  than  the  angle 
DA  C ;  then  we  are  to  prove  that 
the  biase  BC  is  greater  than  the 
base  CD. 

Conceive  the  two  As  joined  together  so  that  the  shorter  sides 
will  be  common  to  them.  As  AB—AD,  ABD  is  an  isosceles  A, 
from  the  vertex  A  draw  a  line  bisecting  the  angle  BAD.  This 
line  must  meet  BC,  and  will  not  meet  CD,  because  the  J  BAC  is 
greater  than  the  _J  DAC,  and  be  perpendicular  to  BD  (th.  15). 
From  U,  where  the  perpendicular  meets  BC,  draw  ED.  ^ 

Now     ....     BE=ED     (th.  15,  scholium).      • 

Add  to  each  EC,  then       BC=ED-{-EC  * 

But  DE-i-EC is  greater  than  DC; 

Therefore         .         .  BCy  DC.     Q.  E,  D. 

• 

THEOREM     20. 

A  perpendicidar  is  the  shortest  line  that  can  he  drawn  from  any  point 
to  a  straight  line  ;  and  if  other  lines  be  drawn  from  the  same  point  to 
the  same  straight  line,  the  greater  vMl  he  at  a  greater  distance  from 
ike  perpendicular  ;  and  lines  at  equal  distances  from  the  perpendicular^ 
en  opposite  sides,  ar6  equcd. 


BOOK    I. 


27 


Let  A  be  any  point  witliout  the 
line  DE;  and  let  AJB  be  the  perpen- 
dicular ;  A  Cy  AD,  and  AJS  oblique 
lines:  then,  if  JBO is  less  than  JSJ), 
and  BC=BE,  we  are  to  show, 

1st.  That  AB  is  less  than  AC. 
2d.  AC  less  than  AT).  3d.  AC=AE. 

In  the  triangle  ABO,  as  AB  is  perpendicular  by  (hy.),  the  angle 
ABC  is  a  right  angle  ;  then,  as  it  requires  the  other  two  angles  of 
the  triangle  (th.  11)  to  make  another  right  angle,  the  angle  ACB, 
is  less  than  a  right  angle  ;  and  as  the  greater  side  is  always  oppo- 
site the  greater  angle,  AB  is  less  than  A  C;  and  as  AO  is  any  line 
differing  from  AB,  therefore  AB  is  the  least  of  any  line  drawn 
from  A, 

2d.  As  the  two  angles  ACB  and  ACB  (th.  1  )  make  two  right 
angles,  and  A  CB  less  than  a  right  angle,  therefore  A  CD  is  greater 
than  a  right  angle ;  consequently,  the  _]  D  is  less  than  a  right 
angle  ;  and,  therefore,  in  the  A  A  CD,  AD  is  greater  than  A  C,  or 
AC  is  less  than  AD. 

3d.  In  the  As  ABC  and  ABU,  AB  is  common,  and  CB=BB, 
and  the  angles  at  B,  right  angles ;  therefore,  by  (th.  15)  ^  C=A£J. 

Q.  U.  D, 


THEOREM     21. 


The  opposite  sides,  and  the  opposite  angles  of  any  parallelogram, 
are  equal  to  each  other. 

Let  ABDC  be  a  parallelogram.  Then  we 
are  to  show  that  AB=CD,  AC=BD,  the  an- 
gle A=D,  and  the  angle  ACD=ABD. 

Draw  a  diagonal,  as  CB ;  then,  because 
AB  and  CD  are  parallel,  the  alternate  an- 
gles ^5(7  and  BCD  (th.  5)  are  equal.  For  the  same  reason,  as 
A  C  and  BD  are  parallel,  the  angles  A  CB  and  CBD  are  equal. 
Now,  in  the  two  triangles  ABCandBCD,  the  side  CB  is  common, 
and 

The  J  ACB=J  CBD    .        .    (1) 
or>H     )BCD=JABO         .        (2) 


28  GEOMETRY. 

Therefore,  the  third  angle  A—  the  third  angle  D  (th.  11),  and  by 
(th.  13)  the  two  As  are  equal  in  all  respects ;  that  is,  the  sides 
opposite  the  equal  angles  are  equal ;  or,  AjB=  CD,  and  A  C—BD. 
By  adding  equations  (1)  and  (2),  (ax.  2),  we  have  the  angle  ^Ci> 
=  the  angle  ABD  ;  therefore,  the  opposite  sides,  &c.     §.  E.  D. 

Cor.  1 .  As  the  sum  of  all  the  angles  of  the  quadrilateral  is 
equal  to  four  right  angles,  and  the  angle  A  is  always  =  to  the 
opposite  angle  D;  if,  therefore,  Ais  o.  right  angle,  D  is  also  a  right 
angle,  and  all  the  angles  are  right  angles. 

Cor.  2.  As  the  angle  ABD,  added  to  the  angle  A,  gives  the 
same  sum  as  the  angles  of  the  £\  ACB ;  therefore,  the  two  ad- 
jacent angles  of  a  parallelogram  make  two  right  angles ;  and  this 
corresponds  with  the  2d  point  of  theorem  12. 


THEOREM    22. 

If  the  opposite  sides  of  a  quadrilateral  are  equal,  they  are  also 
parallel,  and  the  figure  is  a  parallelogram. 

Let  ABDC  represent  any  quadrilateral, 
and  on  the  supposition  that  AC^=BD,  and 
AB=-  CD,  we  are  to  prove  that  AC  is  parallel 
to  BD,  and  AB  parallel  to  CD. 

Draw  the  diagonal  CB  ;  then  we  have  two 
triangles  ABC,  and  CDB,  which  have  the  common  side  CB;  and 
AC  of  the  one=jBZ)  of  the  other,  and  AB  of  the  one=  CD  of  the 
other ;  therefore  by  (th.  19)  the  two  As  are  equal,  and  the  angles 
equal,  to  which  the  equal  sides  are  opposite ;  that  is,  the  angled  CB 
=the  angle  CBD,  and  these  are  alternate  angles  ;  and,  therefore, 
by  (th.  5),  AC  is  parallel  to  BD;  and  because  the  angle  ABC= 
BCD,  AB  is  parallel  to  CD,  and  the  figure  is  a  parallelogram 
Q.  E.  D. 

Cor.  In  this,  and  also  in  (th.  21),  we  proved  that  the  two  As 
which  make  up  the  parallelogram  are  equal ;  and  the  same  would 
be  true  if  we  drew  the  diagonal  from  A  to  D;  and  in  general  we  may 
say,  thai  the  diagonal  of  any  parallelogram  bisects  the  parallelogram. 


BOOK    I.  29 

THEOREM    2  3. 

The  lines  which  join  the  corresponding  extremities  of  two  equal  and 
parallel  straight  lines,  are  themselves  equal  and  parallel ;  and  the 
figure  thus  formed  is  a  parallelogram,. 

On  the  supposition  that  AB  is  equal  and  parallel  to  CD  (see 
last  figure),  we  are  to  show  that  AC  will  he  equal  and  parallel  to  BD  ; 
and  that  will  make  the  figure  a  parallelogram. 

Join  CB;  then  because  AB  and  CD  are  parallel,  and  CB  joins 
them,  the  alternate  angles  ABC  and  BCD  are  equal,  and  the  side 
AB=CD,  and  CB  common  to  the  two  As  ABC  and  CDB ; 
therefore  by  (th.  13)  the  two  triangles  are  equal;  that  is,  AC= 
BD,  the  angle  A=D,  and  A  CB=  CBD;  hence,  AC\%  also  parallel 
to  BD;  and  the  figure  is  a  parallelogram.     Q.  E.  D, 

THEOREM    24. 

Parallelograms  on  the  sam£  base,  and  between  the  same  parallels, 
are  equal  in  surface^ 

Let  ^^^Cand  ABFD  be  two  par- 
allelograms on  the  same  base  AB,  and 
between  the  same  parallel  lines  AB  and 
CD;  then  we  are  to  show  that  these  two 
parallelograms  are  equal. 

Kow  CE  and  FD  are  equal,  because 
they  are  each  equal  to  AB  (th.  21 );  and 
if  from  the  whole  line  (7i>we  take,  in  succession,  (7^  and  .Pi),  there 
will  remain  (ax.  3)  ED=CF;  but  FB=CA.  and  AF=BD 
(th.  21)  ;  hence  we  have  two  As,  CAF  and  FBD,  which  have 
tlie  three  sides  of  the  one  equal  to  the  three  corresponding  sides  of 
the  other,  each  to  each;  and  therefore  by  (th.  19)  the  two  As 
CAF  and  FBD  are  equal.  If  from  the  whole  figure  we  take  away 
the  A  CAF,  the  parallelogram  ABDF  remains  ;  and  if  from  the 
whole  figure  the  other  triangle  FBD  be  taken  away,  the  parallel- 
ogram ABFC  will  remain ;  that  is,  from  the  same  quantity,  if 
equals  are  taken  (ax.  3),  equals  will  be  left ;  or  the  parallelogram 
ABDF=ABFC.     Q.  E,  D. 


M. F 

m 


ad  GEOMETRY. 

THEOREM     25. 

Triangles  on  the  same  base,  and  bettoeen  the  same  parallels^  are  equal 
{in  respect  to  area  or  surface). 

Let  the  two  As  ABE  and  ABF 
have  the  same  base  AB,  and  between 
the  same  parallels  AB  and  CD  ;  then 
we  are  to  show  that  they  are  equal  in 
surface. 

From  B  draw  a  dotted  line,  BD^ 
parallel  to  AF ;  and  from  A  draw  a  dotted  line  A  (7,  parallel  to 
BE ;  and  produce  EF  both  ways,  if  necessary,  to  C  and  D;  then 
the  parallelogram  ABFD=t\\Q  parallelogram  ABCE  (th.  24). 
But  the  A  ABE  is  half  the  parallelogram  ABCE,  and  the  A 
ABF  is  half  the  parallelogram  ABDF;  but  halves  of  equals  are 
equal  (ax.  7) ;  therefore  the  A  ^^^=the  A  ABF,     Q.  E.  D. 

THEOREM    26. 

Parallelograms  on  equal  bases,  and  between  the  sanw  parallels,  are 
equal  in  area. 

Let  ABCJ),  and  EFGff,  be  two  par- 
allelograms on  equal  bases,  AB  and 
EF,  and  between  the  same  parallels  ; 
then  we  are  to  show  that  they  are  equal 
in  area. 

As  AB=EF=JIO ;  but  lines  which  join  equal  and  parallel 
lines,  are  themselves  equal  and  parallel  (th.  23)  ;  therefore,  if  AB 
and  ^6^  be  joined,  the  figure  AB  Gil  is  a  parallelogram = to  ABCD 
(th.  24)  ;  and  if  we  turn  the  whole  figure  over,  the  two  parallel- 
ograms HEFQ  and  RGB  A,  will  stand  on  the  same  base,  HG,  and 
between  the  same  parallels ;  therefore,  HGEF=^HGBA ;  and 
consequently  (ax.  1)  ABCD^EFGH.     Q.  E.  D. 

Cor.  Triangles  on  equal  bases,  and  between  the  same  parallels, 
are  equal ;  for,  join  BD  and  EG,  the  A  ABD  is  half  of  the  par- 
allelogram A  G;  and  the  A  EFG  is  half  of  the  equal  parallelogram 
FH;  therefore,  the   A  ^^i>=the  A  EFG  (ax.  7). 


BOOK    I. 


31 


THEOREM    27. 

If  a  tnangle  and  a  parallelogram  he  upon  the  same  or  equal  bases,  and 
between  the  same  parallels,  the  triangle  wUl  be  half  the  parallelogram. 

Let  ABC  be  a  A,  and  ABLE  a  parallel- 
ogram, on  the  same  base  AB,  and  between 
the  same  parallels ;  then  we  are  to  show  thai 
the  A  ABC  is  half  of  ABDE. 

Draw  the  diagonal  UB  to  the  parallelo- 
gram ;  then,  because  the  two  As  ABC  and  ABE  are  on  the  same 
base,  and  between  the  same  parallels,  they  are  equal  (th.  25)  ;  but 
the  A  ^^^is  half  the  parallelogram  ABDE  (cor.  to  the  22)  ; 
therefore  the  A  ABC  is  half  of  the  same  parallelogram  (ax.  7). 
Q.  E,  i>. 


THEOREM    28. 

The  c&mplementary  parallelograms  of  any  parallelogram  which  are 
abotU  its  diagonal,  are  equal  to  each  other. 

Let  AC  he  Si  parallelogram,  and  BD 
its  diagonal  ;  take  any  point,  as  E,  in 
the  diagonal,  and  from  it  draw  lines 
parallel  to  its  sides  ;  thus  forming  four 
parallelograms. 

We  are  now  to  show  thai  the  comple- 
mentary parallelograms  AE  and  EC,  are  equal. 

By  corollary  to  theorem  22  we  learn  that  the  A  ADB=A 
DBG.  Also  by  the  same  (cor.)  a=6,  and  c=d;  therefore  by 
addition     .         .         .         a-\-c=^b-\-d. 

Now  from  the  whole  AADB  take  the  sum  of  the  two  As 
{a-\-c),  and  from  the  whole  A  DBC  ioke  the  equal  sum  {b-^-d), 
and  the  remainders  AE  and  EC  are  equal  (ax.  3).     Q.  E.  D. 


THEOREM    29. 

The  sides  of  a  parallelogram  vfill  inclose  the  greatest  space  when 
the  angles  are  right  angles. 


32  GEOMETRY. 

Lei  ABDC  be  a  right  angled 
parallelogram,  and  ABha  an  ob- 
lique angled  parallelogram  of  equal 
sides  to  the  other ;  then  we  are  to 
show  that  the  right  angled  'parallelogram  ABDC  is  greaier  than  the 
oth^r,  ABba. 

We  take  Aa=A  C,  Then  Aa  is  less  than  AE,  because  the  per- 
pendicular A  (7,  or  its  equal  Aa^  is  less  than  any  oblique  line  AE 
(th.  20)  ;  therefore  the  line  ab  is  between  the  two  parallels  AB  and 
CF.  The  parallelogram  ABDC=ABFE ;  because  they  are  on 
the  same  base  AB,  and  between  the  same  parallels  (th.  24)  ;  but 
the  parallelogram  ABba  is  but  part  of  the  parallelogram  ABFE ; 
therefore,  ABFE,  or  its  equal  ABDC,  is  greater  than  ABba  ;  but 
the  parallelogram  ABba  has  the  same  length  of  sides,  respectively, 
as  the  parallelogram  ABDC ;  therefore  the  side,  &c.     Q.  E.  D. 

Cor.  It  is  evident,  then,  that  the  area  of  the  parallelogram 
ABba  will  become  less  and  less  as  its  angles  become  more  and 
more  oblique ;  and  greater  and  greater  as  its  angles  become  nearer 
and  nearer  to  right  angles. 

Scholium.  All  parallelograms  (indeed  all  figures)  are  referred 
to  square  units  for  their  measurement,  and  the  unit  may  be  taken  at 
pleasure  ;  it  may  be  an  inch,  a  foot,  a  yard,  a  rod,  a  mile,  &c., 
according  as  convenience  and  propriety  may  dictate.  For  example, 
the  parallelogram  ABD  C  is  measured  by  the  number  of  linear 
units  in  CD,  multiplied  into  the  number  of  linear  units  in  A  C ;  the 
product  will  be  the  square  units  in  ABDC ;  for  conceive  CD  com- 
posed of  any  number  of  equal  parts — say  five — and  each  part  some 
unit  of  linear  measure,  and  AC  composed  of  three  such  units, 
and  from  each  point  of  division  on  CD  draw  ^OHBHII^S 
fines  parallel  U)  AC ;  and  from  each  point  of  IHHHHHI 
division  on  AC  draw  fines  parallel  to  CD  or  ■■HHHHI 
AB  ;  then  it  is  as  obvious  as  an  axiom  that  the  IsSHSBI 
parallelogram  will  contain  5X3=15  square  B^^^^^KfiB 
units ;  and  in  general  the  areas  of  right  angled  parallelograms  are 
found  by  multiplying  the  base  by  the  altitude. 

Right  angled  parallelograms  are  called  rectangles  (def.  26),  and 
the  altitude  of  any  parallelogram,  whether  right  angled  or  not,  is 
the  perpendicular  distance  between  its  opposite  sides. 


BOOK    I. 


33 


THEOREM    30. 

The  area  of  any  plane  triangle  is  measured  by  the  product  of  its  base 
into  half  its  altitude  ;  w  half  the  base  into  the  altitude. 

Let  ABQ  represent  any  triangle,  AB  its 
base,  and  AD  at  right  angles  to  AB  its  alti- 
tude ;  then  we  are  to  show  that  the  area  of  ABO 
is  equal  to  the  product  of  AB  into  one  half  of 
AD  ;  or  the  half  of  AB  into  AD. 

On  AB  construct  the  rectangle  ABED;  and  the  area  of  this 
rectangle  is  measured  by  AB  into  AD  (scholium  to  th.  29)  ;  but 
the  area  of  the  A  ABQ  is  one  half  this  rectangle  (th.  27)  ; 
therefore,  &c.     Q.  E.  D, 


THEOREM    31. 

The  area  of  a  trapezoid  is  measured  by  the  half  sum  of  its  parallel 
sides,  multiplied  into  the  perpendicular  distance  between  them. 

Let  ^^i)  (7  represent  any  trapezoid, 
and  draw  the  diagonal  BO,  which  di- 
vides it  into  two  triangles,  ABG  and 
BCD:  CD  is  the  base  of  one  tri- 
JOigle,  and  AB  may  be  considered  as 
the  base  of  the  other  ;  and  EF  is  the  common  altitude  of  the  two 
triangles. 

Now  by  the  last  theorem  the  area  of  the  triangle  CDB  is=^ 
CDXEF;  and  the  area  of  the  A  ABC=^ABXEF;  therefore, 
by  addition,  the  area  of  the  two  As,  or  of  the  trapezoid,  is  equal 
to  U^B+  CD)  X  EF,     Q.  E.  D, 


THEOREM    32. 

If  there  be  two  lines,  one  of  which  is  divided  into  any  number  of 
parts,  the  rectangle  contained  by  the  two  lines  is  equal  to  the  several 
rectangles  contained  by  the  undivided  line,  and  the  several  parts  of 
(he  divided  line. 


34 


GEOMETRY. 


Let  AB  be  one  line,  and  AD  the  other ; 
and  suppose  AB  divided  into  any  number 
of  parts  at  the  points  U,  F,  G,  &c.  ;  then 
the  whole  rectangle  of  the  two  lines  is  AIT, 
which  is  measured  by  AB  into  AD;  and 
the  rectangle  AL  is  measured  by  AJS  into 
AD;  and  the  rectangle  ^^is  measured  by  JEF  into  EL,  which  is 
equal  to  EF  into  AD;  and  so  of  all  the  other  partial  rectangles  ; 
and  the  truth  of  the  proposition  is  as  obvious  as  that  a  whole  is 
equal  to  the  sum  of  all  its  parts  ;  and  requires  no  other  demon- 
stration than  an  explanation  of  exactly  what  is  meant  by  the  words 
of  the  text. 


THEOREM    33. 

.  If  a  straight  line  he  divided  into  any  two  parts ,  the  sqimre  of  the 
whole  line  is  egtccU  to  the  sum  of  the  squares  of  the  two  parts,  and 
twice  the  rectangle  contained  by  the  parts. 

Let  AB  be  any  line  divided  into  any  two 
parts  at  the  point  0 ;  then  we  are  to  show 
that  the  square  on  AB  is  equal  to  the  sum  of 
the  squares  on  AC  and  CB,  and  twice  the  rec- 
tangle of  AC  into  CB. 

On  AB  describe  the  square  (or  con-, 
ceive    it  described)  AD.     Through   the 
point  C  conceive   CM  drawn  parallel  to 
BD;  and  take  BH=BC;  and  through  H  draw  J522V parallel  to 
ABy  and  CH  is  the  square  on  CBy  by  direct  construction. 

As  AB=BD,  and  CB=Bff,  therefore,  by  subtraction,  AB — 
CB=BD—BH;  or  AC=ffD.  But  NK==ACy  being  opposite 
sides  of  a  parallelogram ;  and  for  the  same  reason  KM^=HD; 
therefore  (ax.  1),  NK=KM ;  and  the  figure  NM  is  a  square  on 
iVX  equal  to  a  square  on  A  0.  But  the  whole  square  on  AB  is  com- 
posed of  the  two  squares  CH,  NM,  and  the  two  complements  or  rec* 
tangles  ^^and  KD;  and  each  of  these  is  ^(7  in  length,  and  BO 
in  width  ;  and  each  has  for  its  measure  A  C  into  CB;  therefore  the 
whole  square  on  AB  is  equal  to  A  C''-\-B  C'-^lA  CX  CB.    Q.  K  D, 

This  may  be  proved  algebraically,  thus  : 


irnmnimnniniiiiiuutMimmffii 


BOOK    I.  35 

"Let  w  represent  any  whole  right  line  divided  into  any  two  parts 
a  and  b;  then  we  shall  have  the  equation 
w=a  -{-b 
By  squaring       w^z=a^-i-b'^-]'2ab,     Q.  E.  D. 

Scholium.     If  a =5^  then  w'=4a^  which  shows  that  the  square 
of  any  whole  line  is  four  times  the  square  of  half  of  it. 


THEOREM     31. 

The  square  on  the  difference  of  two  lines  is  equal  to  the  sum  of  the 
squares  of  the  two  lines,  diminished  by  twice  the  rectangles  contained  by 
the  lines. 

Let  AB  represent  the  greater  line,  -6  (7  a 
lesser  line,  and  A  C  their  difference. 

We  are  now  to  show  that  the  square  on  AG 
is  equal  to  the  sum  of  the  squares  on  AB  and 
B  C,  diminished  by  tvnce  the  rectangle  contained 
by  AB  into  BC. 

On  AB  conceive  the  square  AF  to  be  de- 
scribed ;  and  on    CB  conceive   the   square 
BL  described  ;  and  on  -4(7  describe  the  square  ACGM;  and  pro- 
duce MG  to  K. 

As  aC=A  C,  and  CL=  CB;  therefore,  by  addition,  (  G0-{-  GL), 
or  GL,  is  equal  [AC-{-CB),  or  AB.  Therefore  the  rectangle 
GE  is  AB  in  length,  and  CB  in  width  ;  and  is  measured  by  AB 
into  BC. 

Also  AH'=AB,  and  AM— AC;  therefore  by  subtraction  MH 
=^CB;  and  as  MK=AB,  the  rectangle  UK  is  AB  in  length,  and 
CB  in  width,  and  it  is  measured  by  AB  into  CB;  and  the  two 
rectangles  6^^^  and  JIir,&re  together  equal  to  2ABXBC. 

Now  the  squares  on  AB  and  BC  make  the  whole  figure 
AHFELC ;  and  from  this  whole  figure,  or  these  two  squares,  take 
away  the  two  rectangles  HK  and  GE,  and  the  square  on  ^  (7 
only  will  remain  ;  that  is, 

AC^=AB^-^BC^--2ABXB0.     Q.  E.  D. 
This  may  be  proved  algebraically,  thus: 


36  GEOMETRY. 

Let  a  represent  one  line,  h  another  and  lesser  line,  and  d  their 
difference  ;  then  we  must  have  this  equation : 
d=a — b 
By  squaring     .         .     d^=a^-\-lP-^2ah, 

THEOREM     35. 

The  difference  of  the  squares  of  any  two  lines  is  equal  to  the  rec* 
tangle  contained  hy  the  sum  and  difference  of  the  lines. 

Let  AB  be  one  line,  and  A  C  the  other,  and 
on  them  describe  the  squares  AD,  AM;  then 
the  difference  of  the  squares  on  AB  and  on  AC 
is  the  two  rectangles  UF  and  FC,  We  are  now 
to  show  that  the  measure  of  these  rectangles  may 
be  expressed  hy  (AB-j-AC)  into  (AB — AC). 

The  rectangle  EF  has  ED,  or  its  equal  AB, 
for  its  length  ;  the  other  has  MC,  or  its  equal  A  C,  for  its  length ; 
therefore,  the  two  together  (if  we  conceive  them  put  between  the 
same  parallel  lines)  will  have  {AB-^-AC)  for  the  length;  and 
the  common  width  is  CB,  which  is  equal  to  (AB — A  C);  there- 
fore, AB'--AC^={AB^AC)X{AB—AG).     Q.  E.  D, 

This  is  proved  algebraically  thus  : 

Put  a  to  represent  one  line,  and  b  another  ; 

Then  a-\-b  is  their  sum,  and  a — b  their  difference  ; 
and         .         .    (a'\-b)X(a—i)=a^—b\     Q.  E.  D. 

THEOREM     36. 

The  square  described  on  the  hypotenuse  of  any  right  angled  triangle 
is  equal  to  the  sum  of  the  squares  on  the  other  two  sides. 

Let  ABC  represent  any  right  angled  tri- 
angle, the  right  angle  at  B. 

We  are  to  show  that  the  square  on  AG  is 
equal  to  the  sum  of  two  squares  ;  one  on  AB,  the 
other  on  BC. 

Conceive  the  three  squares,  AD,  AT,  and 
BM,  described  on  the  three  sides.  Through 
the  point  B,  draw  BXE  perpendicular  to  A  (7, 
and  produce  it  to  meet  the  line  6*7  in  K. 

Produce  AF  to  meet  GI  in  H.     If  ML  be 


BOOK    I.  ^ 

produced,  it  will  meet  the  point  K,  and  /^ZJTwill  be  a  right 
angled  parallelogram  ;  for  its  opposite  sides  are  parallel,  and  all  its 
angles  right  angles. 

The  angle  BA  ^  is  a  right  angle,  and  the  angle  XAH  is  also  a 
right  angle  ;  and  from  these  equals  if  we  subtract  the  common 
angle  BAH,  the  remaining  angle,  BA  (7,  must  be  equal  to  the  re- 
maining angle  GAH.  The  angle  ^  is  a  right  angle,  equal  to  the 
angle  ABC;  and  AB=AQ;  therefore,  the  two  As  ABC  and 
AGJI  are  equal,  and  AH=^AC,  But  AG=AF;  therefore  AH 
=AF.  Now  the  two  parallelograms,  AJE  and  ^iT  are  equal,  be- 
cause they  are  upion  equal  bases,  and  between  the  same  parallels, 
FH  and  JEJ^(th.  26). 

But  the  square  AI,  and  the  parallelogram  -4  JST  are  equal,  because 
they  are  on  the  same  base,  AB,  and  between  the  same  parallels, 
AB  and  OIC;  therefore  the  square  AI,  and  the  parallelogram 
AF,  being  both  equal  to  the  same  parallelogram  AF,  are  equal 
to  each  other  (ax.  1 ).  In  the  same  manner  we  may  prove  the 
square  BM  equal  to  the  rectangle  iV"i>/  therefore,  by  addition, 
the  two  squares  Aland  BM,  are  equal  to  the  two  parallelograms 
AF  and  2^D,  or  to  the  square  AD.     Q.  F.  D. 

Scholium.  The  two  sides  AB  and  BC  may  vary,  while  AC 
remains  constant.  ^5  may  be  equal  to  BC ;  then  the  point  iV 
would  be  in  the  middle  of  A  C.  When  AB  is  very  near  the  length 
of  AC,  and  BCyerj  small,  then  the  point  ^V falls  very  near  to  C. 

Now,  as  the  parallelograms  AF  and  JVD  (while  AC  remains 
unchanged)  depend  for  their  relative  magnitudes  on  the  position  of 
the  point  iV,  on  the  line  A  C,  the  area  AF  must  be  to  the  area  JVB 
as  the  line  ^^Y  to  ^C ;  that  is,  the  square  on  AB,  must  be  to  the 
square  on  BC,  as  the  line  AN  to  the  lin£  NC. 

ANOTHER  DEMONSTRATION  OF  THEOREM  36. 

Let  ABC  be  a  right  angled  triangle, 
right  angled  oXA.  Call  AB,  a,  AC,  b,  and 
B  C,  h  :  then  we  are  to  show  that  a^-(-5^=A^. 

Produce  AB  to  D,  making  BD=AG; 
and  produce  A  C  to  F,  making  CF=AB  : 
then  AD^AF;  and  each  of  these  lines  is  (a 
+i),  and  the  whole  square  ^^is  the  square 
of  (a+b),  and  by  (th.  33)  is  a^+b^+2ab. 


88 


GEOMETRY. 


From  B  draw  BG  2ki  right  angles  to  CB  ;  and  from  C  draw  CF 
at  right  angles,  the  same  line  CB ;  then  BG  and  CF  must  be 
parallel,  and  join  FG.  We  must  now  prove  that  the  four  triangles* 
in  the  square  AH  are  all  equal,  and  that  CGBF  is  the  square  on 
CB.  As  the  two  angles  CBA  and  CBD  make  two  right  angles^ 
(th.  1),  and  CBG  is  a  right  angle  by  construction,  therefore  the 
two  angles  CBA  and  GBD  make  one  right  angle.  But  CBA  and 
A  CB  (cor.  4,  th.  11)  are  also  equal  to  a  right  angle  ;  and  from  these 
equals  take  the  angle  CBA,  and  the  angle  GBD  =  the  angle  A  CB, 
But  the  angle  A=  the  angle  D;  both  right  angles,  and  BJ)  was 
made  equal  to  AG;  therefore,  the  two  triangles^  ABC  and  GBJ), 
having  a  side  and  two  angles  equal,  are  in  all  respects  equal,  and 
CB=BG.  In  the  same  manner  we  prove  BG=GF;  and  there- 
fore CG  is  a  square  on  CB,  and  the  four  triangles  are  each  equal 
to  ABC,  and  each  triangle  has  for  its  measure  ^ab.  The  measure 
of  two  of  these  is  ab,  and  the  four  is  2ab. 


Now     . 

Also    . 

By  subtraction 

By  transposition 


AD^=a^-^b^-\-2ab 
AI)^=k^-\-2ab 


0      ■=a^^b'^h^ 

h^     =a^+b\     Q.F,D. 

Cor.  From  this  equation  we  may  have 

A*— a2=5^•or,  (k+a)  {h-a)^I^. 


THEOREM    37. 

In  any  obtuse  angled  triangle  the  square  of  the  side  opposite 
the  obtuse  angle  is  greater  than  the  sum  of  squares  on  the  other  two 
sides,  by  twice  the  rectangle  of  the  base,  and  the  distance  of  the  per" 
pendicular  from  the  obtuse  angle. 

Let  ABC  be  any  obtuse  angled  A,  obtuse 
angled  at  B,  Represent  the  side  opposite  B 
by  b ;  opposite  -4  by  a  /  and  opposite  G  by  c 
(and  let  this  be  a  general  form  of  notation)  : 
also  represent  the  perpendicular  by  p,  and 
J)B  by  X.  Now  we  are  to  show  that  b'^=a^-\' 
c^-\-2ax. 

By(th.  36)     .        .         .    jt>'+(a+ar)2=52 

Also  .         .         .    p'-i'  «'=*c» 


BOOK    I.  89 

By  expanding  equation  (1),  and  subtracting  (2),  we  have 

By  transposition      b^==a^+c^+2ax,     Q.  E,  D. 

S(,nolium.  This  equation  is  true,  whatever  be  the  value  of  x, 
and  X  may  be  of  any  value  less  than  CD.  When  x  is  very  small, 
B  is  very  near  />,  and  the  line  c  is  very  near  in  position  and  value 
to  jt?.  When  a:=0,  c  becomes  j?,  and  the  angle  ABG  becomes  a 
right  angle,  and  the  equation  becomes  IP=a^-{-c^,  corresponding 
to  (th.  36). 


THEOREM     38. 

In  avy  triangle,  the  square  of  a  side  opposite  an  acute  angle  is  less 
than  the  square  of  the  base,  and  the  other  side,  hy  twice  the  rectangle  of 
the  base,  and  the  distance  of  the  perpendicular  from  the  actiie  angle. 

Let  ABC,  eith- 
er figure,  represent 
any  triangle ;  G 
the  acute  angle, 
CB  the  base,  and 
AD  the  perpen- 
dicular, which  falls 
either  without  or  on  the  base.  Then  we  are  to  prove  that  AB* 
^CB'+AC^—^CBX  CD. 

As  in  (th.  37),  put  AB=c,  AC=h,  CB=a,  BD=x,  AD=p; 
and  when  the  perpendicular  falls  without  the  base,  as  in  the  first 
figure,  CD=^a-]-x  ;  when  it  falls  on  the  base,  CD=a — x. 

Considering  the  first  figure,  and  by  the  aid  of  (th.  36),  we  have 
the  following  equations  : 

p'+(a+x)'=b'  (1) 


p'+x''=c' 


(2) 


By  expanding  (I),  and  subtracting  (2),  we  have 

a^-i-2aj:=b^—c^ 
By  adding  a^  to  both  members,  and  transposing  c',  we  have 

c'-{-(2a^-\-2ax)=b^-\-a^ 
By  transposing    the   vinculum,  and  resolving  it  into   factors, 
^^c  have 

c':=a'+b'--2a(a+x).     Q.  K  D. 


40  GEOMETRY. 

Considering  the  other  figure,  we  have 


(1) 

(2) 


By  subtraction  a^ — 2ax        =b^ — c^ 

By  adding  a^  to  both  members,  and  transposing  c^  we  have 

.      c^=b'^'\-a'—2a(a—x).     Q.  E.  D. 


THEOREM     39. 

If  in  any  triangle  a  line  be  drawn  from  any  angle  to  the  middle  of 
the  opposite  side,  tmce  the  square  of  this  lin£,  together  with  twice  the 
square  of  half  the  side  bisected,  will  be  equal  to  the  sum  of  the  squares 
of  the  other  two  sides* 

Let  ABC  be  a  triangle,  its  base 
bisected  in  M.  Tlien  we  are  to  prove 
that  2AM2+2CM2=AC2+AB^ 

Draw  AD  perpendicular  to  the 
base,  and  call  it  p.  Put  AC=:by 
AjB=c,  CB=2a ;  then  CM=:a,  and 
MB=a.  Make  MD=x ;  then  CD 
=za-{-x,  and  DB=a — x.    Put  AM=m, 

Now  by  (th.  36)  we  have  the  two  following  equations  : 
^24-(a— a;)2=c2  (1) 

'  p^-\-(a^xy=b^  (2) 

By  addition     .  2p2+2ar2-f-2a2=62+c^     ^viip''-^x'=m 
Therefore  2m''-\-2a''=b''^-c\  Q.  E.  D. 


THEOREM    40. 

The  two  diagonals  of  any  parallelogram  bisect  each  other;  and 
the  sum  of  their  squares  is  equal  to  the  sum  of  the  squares  of  all  the 
four  sides  of  the  parallelogram. 

Let  ABCD  be  any  parallelogram,  and 
draw  its  diagonals  A  C  and  BD, 

We  are  now  to  show,  1st.  Thai  AE 
=EC,  DE=EB.  2d.  That  AC^+BD^ 
=AB2+BC=»+DG='+AD^ 


BOOK    I.  41 

1.  The  two  triangles  ABB  and  BBC  are  equal,  because  AB 
xnDC,  the  angle  ABE  —  the  alternate  angle  EDO,  and  the  vertical 
angles  at  E  are  equal ;  therefor*^,  AE,  the  side  opposite  the  angle 
ABE,  is  equal  to  EC,  the  side  opposite  the  equal  angle  EDC : 
also  EB,  the  remaining  side  of  the  one  A  is  equal  to  ED,  the 
remaining  side  of  the  other  triangle. 

2.  As  ADC'\?>  a  triangle  whor-e  base  AC  \%  bisected  in  E,  we 
have,  by  (th.  39), 

<iAE^-\-<lED'=^AD''^DC''        ( 1 ) 
As  ^^6'  is  a  triangle  whose  base,  A  C,  is  bisected  in  E,  we  have 

^AE''^'iEB^=AB^-\-BC^  (2) 

By  adding  equations  (1)  and  (2),  and  observing  that 
EB^=^ED^,  we  have 
AAE^-\-4.ED^=AD^-\-DC^-\-AB^-\-BC^ 
But  four  times  the  square  of  the  half  of  a  line  is  equal  to  the 
square  of  the  whole  (scholium  to  th.  33);  therefore  4AE^=AC\ 
and  4ED^=DB^ ;  and  by  making  the  substitutions  we  have 

AO'-^-DB'^AD'+DO'+AB'-^BC^.     Q.  E.  D. 


49  GEOMETRY 


BOOK     II 


PEOPORTION. 

The  word  Proportion  has  different  shades  of  meaning,  accord* 
ing  to  the  subject  to  which  it  is  applied :  thus,  when  we  say  that  a 
person,  a  building,  or  a  vessel  is  ytqW  proportioned,  we  mean  nothing 
more  than  that  the  different  parts  of  the  person  or  thing  bear  that 
general  relation  to  each  other  which  corresponds  to  our  taste  and 
ideas  of  beauty  or  utility,  but  in  a  more  concise  and  geometrical 
sense, 

Proportion  is  the  numerical  relation  which  one  quantity  hears  to 
another  of  the  same  kind, 

DEFINITIONS  AND  EXPLANATIONS. 

In  Geometry,  the  quantities  between  which  proportion  can 
exist,  are  of  three  kinds,  only.  1st.  A  line  to  a  line,  2d.  A  sur- 
face to  a  surface.     3d.  A  solid  to  a  solid. 

To  find  the  numerical  relation  which  one  quantity  bears  to 
another,  we  must  refer  them  both  to  the  same  standard  of  measure. 

If  a  quantity,  as  A,  be  contained  exactly        A 
a  certain  number  of  times  in  another  quan-        „ 

tity,  J5,  the  quantity  A  is  said  to  measure     i — i 1 » 

the  quantity  B;  and  if  the  same  quantity,      ^ |_^_j ^ 

.^,  be  contained  exactly  a  certain  number 

of  times  in  another  quantity,  (7,  A  is  also     | 1 

said  to  be  a  measure  of  the  quantity  (7,  and        JS 
it  is  called  a  common  measure  of  the  quan-  ^ 

titles -B  and  G;  and  the  quantities  B  and     ' ' '        '        ' 

C  will,  evidently,  bear  the  same  relation  to  each  other  that  the 
numbers  do  which  represent  the  multiple  that  each  quantity  is  of 
the  common  measure  A. 

Thus,  if  B  contain  A  three  times,  and  C  contain  A  also  three 
times,  B  and  C  being  equimultiples  of  the  quantity  A^  will  be 


BOOK   II.  4a 

equal  to  each  other  ;  and  if  'B  contain  A  three  times,  and  C  con- 
tain A  four  times,  the  proportion  between  B  and  C  will  be  the 
same  as  the  proportion  between  the  numbers  3  and  4. 

Again,  if  a  quantity,  -D,  be  contained  as  often  in  another  quan- 
tity, JEf  as  A  is  contained  in  Bt  and  as  often  in  another  quantity, 
-F,  as  -4  is  contained  in  (7,  the  ratio  of  E  to  F,  or  the  proportion 
between  them,  will  be  the  same  as  the  proportion  between  B  and 
C;  and  in  that  case,  the  quantities  B,  C,  E,  and  F,  are  said  to  be 
proportional  quantities ;  a  relation  which  is  commonly  expressed 
thus,  B\  CwE'.F. 

To  find  the  numerical  relation  that  any  quantity,  as  A,  has  to 
any  other  quantity  of  the  same  kind  as  By  we  simply  divide  B  by 
Ay  and  the  quotient  may  appear  in  the  form  of  a  fraction,  thus : 

■n 

^.    Now  this  fraction,  or  the  value  of  this  quotient,  is  always  a 

numeral^  whatever  quantities  may  be  expressed  by  A  and  B. 
To  find  the  numerical  relation  between  D  and  Ey  we  simply 

divide  I)  by  Ey  or  write  -  ,  which  denotes  the  division ;  and  if  we 
E 

find  the  same  quotient  as  when  we  divided  B  by  -4,  then  we  may 
write 

A    E        ^  ' 

If  B  contains  A  three  times,  and  D  contains  E  three  times,  as 
we  have  just  supposed,  equation  ( 1 )  is  nothing  more  than  saying 
that 

3=3 

When  we  divide  one  quantity  by  another  to  find  their  numerical 
relation,  the  quotient  thus  obtained  is  called  the  ratio. 

When  the  ratio  between  two  qtiantiiies  is  the  same  as  the  ratio  between 
two  other  quantities  y  the  four  quantities  constituie  a  proportion. 

N.  B.  On  this  single  definition  rests  the  whole  subject  of  geo- 
metrical proportion. 

On  this  definition,  if  we  suppose  that  B  is  any  number  of  times 
Ay  and  D  the  same  number  of  times  E,  then 
-4  is  to  J5  as  -£'  is  to  D; 
Or  more  concisely : 

A :  B=E:  D,     The  signs  :  =  :  meaning  equal  ratio. 


44  GEOMETRY. 

Now  it  is  manifest,  that  if  E  is  greater  than  A,  D  will  be  greater 
than  B.  If  A=Ey  then  JB=I>,  ckc,  &c. ;  and  whatever  relation 
or  ratio  A  is  of  E,  the  same  ratio  B  will  be  of  D;  and  whatever 
relation  j5  is  of  A,  the  same  relation  D  will  be  of  E.  This  shows 
that  the  means  may  be  changed,  or  made  to  change  places. 

Or,  .  .  .  A:  E=B  :  i),  which  is  the  former  pro- 
portion  with  the  middle  terms  or  means  changed. 

The  Jirst  and  third  of  four  magnitudes  are  called  the  antecedents ; 
the  second  and  fourth,  the  consequents. 

A  simple  relation  or  ratio  exists  between  any  two  magnitudes  of 
the  same  kind  ;  but  a  proportion,  in  the  full  sense  of  tlie  term, 
must  consist  of  four  quantities. 

When  the  two  middle  quantities  are  equal,  as, 

A:B=B:0 
then  the  three  quantities.  A,  B,  and  C,  are  said  to  be  continued 
proportionals  ;  and  B  is  said  to  be  the  mean  proportional  between 
A  and  C;  and  C  is  said  to  be  the  third  proportional  to  A  and  B. 

In  the  proportion  A :  B=  C :  J),  the  last  D  is  said  to  be  the 
fourth  proportional  to  AylB,  and  0. 

By  the  same  rule  of  expression,  A  may  be  called  the  first  pro- 
portional, B  the  second,  and  C  the  third ;  for  either  one  can  be 
found  when  the  other  three  are  given,  as  we  shall  subsequently 
explain.. 

When  quantities  have  the  same  constant  ratio  from  one  to  the 
other,  they  are  said  to  be  in  continued  proportion, 

Thus:  the  numbers  1,  2,  4,  8,  16,  <kc.,  are  in  continued  pro* 
portion  ;  the  constant  ratio  from  term  to  term  being  2. 

THEOREM     1. 

Jf  there  be  two  magnitudes  which  have  a  comnwn  m£asure,  x,  so 
thai  the  first  magnitude  mmj  he  expressed  by  mx,  the  second  by  nx ;  and 
two  other  magnitudes  which  have  a  common  measure,  y,  so  that  the 
first  mm/  be  expressed  by  my,  the  second  by  ny ;  that  is,  the  two  com- 
mon measures  x  and  y  having  the  same  equimultiples,  m  and  n,  to 
make  up  the  magnitudes  ;  then  the /our  magnitudes  vMl  be  in  geomc" 
triced  proportion. 

Or        .        .        .    ma:nx^=my:ny 


BOOK    II.  45 

For  the  ratio  between  mx  and  nxis   —  =— ,  and  the  ratio  between 

7nx    m 

my  and  wy  is   —  z=  — ,  the  same  ratio;  therefore,  by  the  definition 
my     m 

of  proportion,  these  magnitudes  are  proportional.     Q.  E,  D. 

Scholium.  If  we  change  the  means,  the  magnitudes  are  still 
proportional ;  but  the  ratio  between  the  terms  of  comparison  is 
different. 

Thus;        .        .        mx:my:=inx:ny. 

The  ratio  between  the  1st  and  2d,  is,  -i-=?;  the  ratio  between 

mx    X 

the  3d  and  4th  is  !^=?l,  the  same  ratio  as  between  the  other  two 

nx    X 
magnitudes  ;  but  as  in  this  latter  case  we  compare  diflferent  mag- 
nitudes, the  numeral  value  of  the  ratio  is  different. 

But  we  cannot  change  the  means,  unless  we  then  consider  the 
magnitudes  existing  only  in  their  numeral  relations.  To  whatever 
the  magnitudes  may  refer,  whether  to  lines,  surfaces,  or  solids,  the 
ratio  is  always  a  mere  numeral ;  therefore,  when  two  ratios  stand 
equal,  we  may  increase  or  decrease  them  at  pleasure,  as  will  be 
shown  hereafter. 

N.  B.  The  first  two  terms  of  a  proportion  are  called  the  first 
couplet,  and  the  last  two  are  called  the  second  couplet. 

THEOREM     2. 

When  four  magnitudes  are  in  geometrical  proportion,  the  product  o/ 
the  extremes  is  equal  to  the  product  of  the  means. 

Let  the  four  magnitudes  be  represented  by  A,  B,  C,  and  D. 

Then        .        .        .   A:B=C:I>. 

Some  numeral  relation,  or  ratio,  must  exist  between  A  and  B. 
Let  that  ratio  be  represented  by  r;  that  is,  B  must  equal  rA. 

But,  by  the  definition  of  proportion,  the  same  relation  must  exist 
between  Cand  D  as  between  A  and  B;  or  D=rC. 

Then  by  substitution  we  have 

A:rA==C:rC. 

The  product  of  the  extremes  is  rCA,  and  that  of  the  means  is 
ArC;  obviously  the  same.     Q.  E.  D, 


46  GEOMETRY. 

THEOREM     3. 

If  three  magnitudes  he  continued  proportionals y  the  product  of  the 
extremes  is  equal  to  the  squxire  of  the  mean. 

Let  Ay  By  and  C  represent  the  three  magnitudes  : 

Then     .         .A:  B=B :  C,  by  the  definition  of  proportion. 

But  by  theorem  2  (book  2),  the  product  of  the  extremes  is  equal 
to  the  product  of  the  means  ;  that  is,  -4X  (7=ji5^.     Q.  E.  D. 

THEOREM    4. 

Equimultiples  of  any  two  magnitudes  have  the  same  ratio  as  the 
magnitudes  themselves ;  and  the  magnitudes  and  their  equimultiples 
wxiy  therefore  form  a  proportion. 

Let  A  and  B  represent  the  magnitudes,  and  mA  and  mB  their 
equimultiples. 

Then    .        .        .    A'.B=mA:mB 

7?  •>»  7?        H 

For  the  ratio  of  -4  to  ^  is  _,  and  of  mA  to  mB  is  —  =— ,   the 

A  mA     A 

same  ratio ;  therefore,  <fec.     Q.  E.  D. 

THEOREM   5. 

If  four  quantities  he  proportional y  they  will  he  proportional  when 
taken  inversely. 

If  A  :  B=mA  :  mB,  then  B  :  A=mB  :  mA  ; 

For  in  either  case,  the  product  of  the  extremes  and  means  are 
manifestly  equal ;  or  the  ratio  between  the  couplets  is  the  same  ; 
therefore,  &c.     Q.  E.  D. 

THEOREM    6. 

Magnitudes  which  are  proportional  to  the  same  proportionals y  art 
proportional  to  each  other. 

If  ,    A'.  B—P  :  Q I       Then  we  are  to  prove  that 

and       .      a:h=F:QS  A:B=a:h. 

By  the  law  of  proportion  -j  =  p- 

h     Q 
Also   .        .        «        •      -=Tr 

a     P 


Let 

.     A:B=C:D] 

And       . 

.     0:D=E:F 

And       . 

,    E\F^G'.H 

&c.=&c. 

BOOK    n.  47 

7?      h 
Therefore,  by  (ax.  1)         — =-,  or  A  :  J?=a  \h     Q.  K  D. 

Cor,  This  principle  may  be  extended  through  any  number  of 
proportionals. 

THEOREM    7. 

^  any  number  of  quantities  be  proportional,  then  any  one  of  the 
antecedents  will  be  to  its  consequent  as  the  sum  of  all  the  ayiiecederd^  is 
to  the  sum  of  all  the  consequents. 


(1) 


Then  we  are  to  show  that 

A  :  B-=  (7+^+  O  &c. :  D-\-F-\-H,  &c. 
If  ^  :  ^  as  C  :  i),  then  some  factor,  whole  or  fractional,  multi- 
plied by  A,  will  produce  C ;  and  the  same  factor  multiplied  by  B, 
will  produce  D;  that  is,  the  proportions  ( 1 )  become 
A  :  B=mA  :  mB 
=  nA  :  nB 
=  pA  :pB 
&c.,    &c. 
But,         A  :  B=mA-^nA-{-pA,  &c  :  mB-{-nB-{-pB,  ckc. 

„      ,  .  B      (m-^n+p)B 

For  the  ratio   .      .      -:  =\     ,      \^{  ^ 

A      {mirn-f-pjA 

Now  as    .        .      .    mA=0,  nA=By  pA=G,  ^c. 

Therefore,       .      A:B=C-\-i;-{-G :  d'-{-F+B.    .    Q.  F.  D. 


THEOREM    8. 

If  four  magnitudes  constitute  a  proportion,  the  first  will  be  to  the 
sum  of  the  fir  si  and  second,  as  the  third  is  to  the  sum  of  the  third  and 
fourth. 

By  hypothesis,  A\B\'.G\D;  then  we  are  to  prove  that 
A'.A^-B'.\  C:  C-\-D, 

T.     1.      .  .        B    D 

Uy  the  given  proportion,   "T^TJ* 


48  GEOMETRY. 

Add  unity  to  both  members,  and  reducing  them  to  the  form  ol 

a  fraction,  we  have  — -: — = — 5-—.     Throwing  this  equation  into 

its  equivalent  proportional  form,  we  have 

A:A-\-£::  C:C-\-D. 
N.  B.     In  place  of  adding  unity,  subtract  it,  and  we  shall  find 
that 

A:A---B::C:C—D 
Or        .        .        A:B—A::C:D^a 

THEOREM    9. 

If  four  magnitudes  he  proportional,  the  sum  of  the  first  aitd  second 
is  to  their  difference,  as  the  sum  of  the  third  and  fourth  is  to  their 
difference. 

Admitting  that      .      A  :  B  :  :  C '.  D,  ytq  slyq  to  prove  that 

A-^B :  A^B  : :  C-{-D  :  (7— i) 
From  the  same  hypothesis,    th.  8    gives 
A'.A^-B'.:  G\  C^D 
And        .        .     A\A—B\\C\C—D 

Changing  the  means  (which  will  not  affect  the  product  of  th< 
extremes  and  means,  and  of  course  will  not  destroy  proportionality), 
and  we  have 

A\  C\\A^-B\  C^B 
A:  C::A—B:  C—D 
Now,  by  (th.  2),        A-\-B  :  C^-D  :  :  A—B  :  C^-J) 
Changing  the  means,  A-\-B  :  A^B  :  :  O+B  :  C—D 

THEOREM     10. 

If  four  magnitudes  he  proportional,  like  powers  or  roots  of  the 

game  will  be  proportional. 

Admitting        .        A  :  B  :  :  C :  B,  we  are  to  show  that 

L      '         LI 

A'' :  jS"  : :  (7« :  i>»,  and  ^":  ^": :  c":  J>* 

A      0 
By  the  hypothesis,    —  =  y-.      Raising    both  members   of    this 

equation  to  the  nth  power,  and 


BOOK    II.  49 


Changing  this  to  the  proportion  A^  :  B*::  C*:  JD* 

B    D' 


A     G 
Resuming  again  the  equation  -D="n»  ^^'^  taking  the  nth  root 


-4*       C" 
of  each  member,  we  have    — j— ==— j-.    Converting  this  equa- 

B        D 

tion  into  its  equivalent  proportion,  we  have 
I      i         L      L 

»        n  n         n 

A\  B  ::(7  :i> 

Now  by  the  first  part  of  this  theorem,  we  have 

mm  5         2 

n  n  n  n 

A  \B    \\C   \  D       w*  representing  any 
power  whatever,  and  n  representing  any  root. 

THEOREM     11. 

If  four  magnitudes  he  proportional,  also  four  others ^  their  com- 
pound,  or  product  of  term  hy  term,  vMlform  a  proportion. 

Admitting  that    .        A'.   B:\      O    :D 
And    .        .        .        X:    V::     M  :  N 
We  are  to  show  that  AX:  BY: ;  MG   :  ND 

A     G 
From  the  first  proportion,   ^-=7^ 

From  the  second,  ~v^^l^ 

Multiply  these  equations,  member  by  member,  and 

AX^MG 

BY    ND 
Or        .        .  AX:BY'.:MG:ND 

The  same  would  be  true  in  any  number  of  proportions. 

THEOREM    12. 

Taking  the  same  hypothesis  as  in  (th.ll),  we  propose  to  show,  that 
a  proportion  may  be  formed  by  dividing  one  proportion  by  the  other, 
term  by  term. 

By  hypothesis,        .  A  :  B  :  :  G  :  D 

And       .        .        .  X:Y:iM:2f 

4 


50 


\                                         GEOMETRY. 

Multiply  extremes  and  means,         AD'=BC 

(1) 

And J!fX=Mr 

(2) 

Divide  (1)  by  (2),  and  .          d.x^=Jx  J 

Convert  these  four  terms,  which  make  two  equal  products,  into 
a  proportion,  and  we  shall  have 

By  comparing  this  with  the  given  proportions,  we  find  it  com- 
posed of  the  quotients  of  the  several  terms  of  the  first  proportion, 
divided  by  the  corresponding  term  of  the  second. 

THEOREM     13. 

Iffov,r  magnitudes  he  proportional ^  we  may  multiply  the  first  couplet 
or  the  second  couplety  the  antecedents  or  the  consequents,  or  divide  them 
by  the  same  factor,  and  the  results  will  he  proportional  in  every  case. 

Suppose        .        .        .        .  A  :  B :  :  C:  J) 

Multiply  extremes  and  means,  and        AI>=JBO        (1) 
Multiply  this  equation  by  J/,  and      MAD=MBC 
Now,  in  this  last  equation,  MA  may  be  considered  as  a  single 
term  or  factor,  or  MD  may  be  so  considered.     So,  in  the  second 
member,  we  may  take  3fB  as  one  factor,  or  MC.     Hence,  we  may 
eonvert  this  equation  into  a  proportion  in  four  different  ways. 


Thus,  as    . 

•        JuA  I  juB  '. 

C 

.D 

Or  as 

A     :B     :: 

MC 

MD 

Or  as 

MAiB     : 

'.MC 

.J) 

Or  as 

A      :MB: 

C 

'.MD 

If  we  resume  the  original  equation  (1),  and  divide  it  by  any 
number,  M,  in  place  of  multiplying  it,  we  can  have,  by  the  same 
•ourse  of  reasoning. 


A    B  . 
M'  M' 

:  C:D 

A:B  : 

C    D 
'MM 

i-': 

4- 

.4. 

-4 

BOOK    II.  51 


THEOREM     14. 

If  three  magnitiules  are  in  continued  proportion,  the  first  m  to  the 
third,  as  the  sqitare  of  the  first  is  to  the  square  of  the  second. 

Let  w4,  J5,  and  (7,  represent  three  proportionals. 

Then  we  are  to  show  that    A  :  0=A^  :  B* 

By  (th.  3)  AC^B" 

Multiply  this  equation  by  the  numeral  value  of  A,  then  we  hav§ 

A^C^AB" 

This  equation  gives  the  following  proportion : 

A  :  (7=^'  :  B\         §.  E.  D, 


THEOREM     15. 

If  any  one  of  the  four  mar/mtudes  which  fomi  a  proportion^  be 
effaced  or  unknown,  it  can  he  rentored  by  means  of  the  other  three. 

Let  A  :  B=C  :  D  represent  a  proportion,  and  suppose  D  un- 
known ;  then  represent  it  by  ^r 

That  is         .         .         A\  B=C  :  x 

The  ratio  between  A  and  B  is  the  same  as  between  C  and  x. 

Represent  the  ratio  between  A  and  B  hy  r;  and  as  r  is  always 
a  numeral,  whatever    quantities   are  represented  by  A  and  B, 

X 

therefore,   p^=r;  or  x-=rC ;  which  shows  that  x  or  D  must  be  of 

the  same  name  as  C 

When  A  and  B  are  not  commensurable,  the  ratio  is  expressed 

.       B       ^        CB         , 

by    ~j  and  rc=— -;  or,  m  numbers,  the  product  of  the  second  and 

third  terms  divided  by  the  first,  will  give  the  fourth,  which  is  the 
rtde  of  three  in  arithmetic. 
In  short,  as 

An     Tin     A     ^^      j^     ^^     n    ^^        4  n     ^^ 
AD^BC,  ^=-^-,    -5=-^.   ^^~B~*  *^^-^=^- 


52 


GEOME  TRY 


THEOREM     16. 

Parallelograms,  and  also  triangles,  having  the  same  or  equal  atti- 
tudes, are  to  one  another  as  their  bases. 

Let  a  represent  the  number  of  units, 
and  part  of  a  unit  in  BC,  and  h  the  num- 
ber of  units  and  part  of  a  unit  in  BD. 

Also  let^  represent  the  units  and  parts 
of  a  imit  in  the  perpendicular,  AB.  Now  by  (scholium  to  th.  29 
book  1),  the  parallelogram  ABCE=pa,  and  the  parallelogram 
ABDF=ph ;  and  as  magnitudes  must  be  proportional  to 
themselves, 

ABCE  :  ABDF=pa  :  pb 

But        .         .  a  :  b=pa  :  pb      (th.  4  book  2) 

Therefore  (th.  6  book  2),  we  have 

ABCE  :  ABDF=a  :  b.  Q.  E,  I). 

Car  1.  As  triangles  on  the  same  base  and  altitude  as  parallel- 
ograms are  halves  of  parallelograms ;  and  as  the  halves  of  quan- 
tities are  in  the  same  proportion  as  their  wholes  ;  therefore 

The         .         .     ABPC  :  A  BQD=a  :  b. 

Cot.  2.  When  parallelograms  and  triangles  have  the  same  or 
equal  basis,  they  will  be  to  each  other  as  their  altitudes  ;  for  the 
proportion  ABCE  \  ABDF=pa\pb,  as  above,  is  always  true; 
and  when  a  becomes  equal  to  b  and  p,  and  p  different, 

Then        .         .        ABCE:  ABDF=Fa:pa 

Or  .         ,        ABCE:ABDF-^P   :p,  that  is,  as  their 

perpendicular  altitudes. 

THEOREM     17. 

Lines  drawn  parallel  to  the  base  of  a  triangle,  ctU  the  sides  of  the 
triangle  proportionally. 

Let  ABC  be  any  triangle,  and 
draw  2)-£^  parallel  to  the  base  BG; 
then  we  are  to  show  thai 

AD  :  BB=AE  :  EC. 

Join  DC  and  BE.  The  triangle 
DEB  =  the  A  DEC,  because  they 
are  on  the  same  base,  DE,  and  be- 
tween the  same  parallels,  DE  and  -B(7  (th.  25  book  1). 


BOOK    II. 


iS 


Represent   the  triangle  ADE  by  T,  DEB  by  x,  DEC  by  y; 

then  x=y.     Now,  as  the  triangles  T'and  x  may  be  considered  as 

having  AD  and  DB  for  bases,  and  the  perpendicular  distance  of 

the  point  E  from  AB  for  altitudes,  therefore,  by  (th.  16,  book  2). 

AD:DB==T:x 

By  reasoning  in  the  same  manner  in  reference  to  the  triangles 
T  and  y,  they  having  their  common  vertex  in  jO,  we  have  the 
proportion 

AE  :  EC=T\y. 

AE\EC=T 

AD:DB=:T 


Therefore 
But 


-A 


But  x=y 

Therefore,  (th.  6,  book  2) 
AE :  EC^AD  :  DB 
Or  AD:  DB=AE  :  EG, 

Q.  E.  D, 

Cur.     Considering  AEB   as   one  triangle,  and  AED  another, 
having  their  common  vertex  mE;  and  in  the  same  manner,  ADC 
as  one,  and  ADE  another,  whose  vertex  is  i>,  then  we  may  have 
AB  : AD=AG :  AE 
For,  by  taking  the  proportion 

AD : DB=AE : EC 
And  by  composition,  (th.  8  book  2),  we  have 
AB  :  AD=AC  :AE. 


THEOREM     18. 

Similar  trianyles  have  their  sides,  about  the  equal  angles^ 
pro}Jorti<mal. 

IsQiABCoxiADEEhQ 
two  similar  triangles, 
having  the  angle  A=D, 
B^-E,  and  C=^F ;  and 
for  the  sake  of  perspi- 
cuity, we  will  suppose 
AB  greater  than  ED. 
Now  we  are  to  show  that  AB  :  A  C=DE  :  DF ;  or  that 
AB  :  DE=AG:  DF. 

Conceive  the  triangle  DEF  taken  up  and  placed  on  the  triangle 
ABC,  in  such  a  manner  that  the  point  D  shall  fall  on  A,  and  the 


54  GEOMETRY. 

line  DE  on  AB,  the  point  E  falling  on  H.  Now,  as  the  angle 
E=B,  the  line  EF,  or  its  representatiye,  Hly  will  take  the  direc- 
tion of  BC,  and  be  parallel  to  BC  (def.  of  parallel  lines). 

Now  the  two  triangles  DEE  and  ABI  are  identical ;  for 
AH=DE,  and  A^D,  and  AHI=E;  then  AIH=F;  therefore 
AI—BF,  and  HI=EF.     But  as  i37  is  parallel  to  jBC,  by  the 

last  theorem  we  have 

AB  :  AC=zAH :  Al 

That  is,        .        ,       AB\A  C=DE  :  DF        Q.  E,  J). 

Scholium,  If  perpendiculars  be  let  fall  from  like  angles  in  the 
triangles,  to  the  opposite  sides,  as  CL  and  EM,  such  perpendiculars 
will  divide  the  two  triangles  into  similar  partial  triangles,  and 

As        .        .        .      AB:J)E=AC:DE 

And      .        .        .       CL:ME=-AO:DE 

Therefore  (th.  6  b.  2)  AB  ;  J)E=  CL  :  AlF 

THEOREM      19. 

-5^  any  triangle  have  its  sides  respectively/  proportional  to  the  like 
tides  of  another  triangle,  each  to  each,  then  the  two  triangles  will  be 
equiangzdar. 

Let  the  triangle  ahc  have  its  sides  proportional 
to  the  triangle  ABO;  that  is,  ac  to  -4(7,  as  cb  to 
CB,  and  ac  to  A  0,  as  ab  to  AB  ;  then  we  are  to 
prove  that  the  A  obc  is  equian- 
gular to  the  A  ABC. 

On  the  other  side  of  the  base, 
AB,  and  from  A,  conceive  the 
angle  BAD  to  be  drawn  =  to  the 
angle  a;  and  from  the  point  B, 
conceive  the  angle  ABD  drawn 
=  to  the  J  b.  Then  the  third  J 
=  to  the  third  angle  C  (th.  11,  cor.  2,  b.  1)  ;  and  the  A  ABD 
will  be  equiangular  to  the  A  cibc  by  construction. 

Therefore,        .         ,    ac  \    ah=AD  :  AB 

By  hypothesis,         .     ac  :    ab=A  C  :  AB 

Hence,     .         .         AD  :AB=AC  :  AB        (th.  6,  b.  2). 

In  this  last  proportion  the  consequents  are  equal ;  therefore,  the 
antecedents  are  equal :  that  is,  AD=AC 

In  the  same  manner  we  prove  that  BD=  CB 


BOOK    II 


55 


iut  AB  is  common  to  the  two  triangles  ;  therefore,  all  three  of 
he  sides  of  the  A  ABD  are  respectively  equal  to  all  three  of  the 
sides  of  the  A  ABC  (th.  19,  b.  1). 

But  the  A  ABD  is  equiangular  to  the  A  ohc  by  construction ; 
therefore,  the  A  ABC  is  also  equiangular  to  the  A  ahc.     Q.  E.  D, 

THEOREM    20. 

Jff^  two  triangles  have  one  angle  in  the  one  equal  to  one  angle  in  the 
other,  and  the  sides  dboid  these  equal  angles,  directly,  or  reciprocally 
'proportional,  the  two  triangles  mil  be  equiangular. 

Let  ABC  and  ahc  be  two  As,  and  the  angle 
A=a,  and  AC  oi  the  one  to  ac  of  the  other,  as 
^^  to  ab.  Then  we  are  to  show  that  the  angle 
B=b,  and  the  angle  c=C. 

If  we  take  the  A  ohc,  turn  it  over  and  place 
the  point  a  on  A,  ac  on  A  C,  and  ah  on  AB, 
and  join  cb,  then  cb  will  be  parallel  to  CB ; 
for  if  c5  be  not  parallel  to  CB,  draw  en  par- 
allel to  CB. 

Then  AC  '.  AB  :  :  An  :  Ac  (th.  17,  b.  2) 

Also   AC :  AB  :  :  Ab  :  Ac    (hy.) 

Now  as  three  terms  in  each  of  these 
proportions  are  the  same,  the  other  terms 
must  be  equal :  that  is,  Ab=An,  and  cb 
and  en  is  the  same  line.  But  en  was  drawn  parallel  to  CB; 
that  is,  cb  is  parallel  to  CB;  therefore,  the  angle  Cz=c  by  the  defi- 
nition of  parallel  lines.     Therefore,  &c.     Q.  E.  D. 

THEOREM     21. 

When  four  straight  lines  are  in  proportion,  the  product  of  the 
extremes  is  equal  to  the  product  of  the  means.* 

Let  A,  B,  C,  D,  represent  the  four  lines  A  i 1 

B  1 . 

Then  we  are  to  show,  geometrically,  thai  C  i ^i 

A'D==C'B.  D  ' 1 


/ 

\ 

/ 

1 .. 

y^B 

r 

^ 
/ 

/ 

/>" 

i^ 

a 

*  This  proposition  has  had  a  symbolical  proof,  in  theorem  2  book  2,  but  wo 
deem  it  important  to  give  this  geometrical  demonstration. 


56 


GEOMETRY. 


Place  A  and  B  at  right  angles  with  each 
other,  and  draw  the  hypotenuse.  Also  place  C 
and  D  at  right  angles  to  each  other,  and  draw 
its  hypotenuse.  Then  bring  the  two  triangles 
together,  so  that  C  «hall  be  at  right  angles  with 
jB,  as  represented  in  the  figure. 

Now,  these  two  As  have  each  a  right  j ,  and 
the  sides  about  the  equal  angles,  proportional ; 
that  is,  A  :  B=C  :  D;  therefore,  (th.  20,  b.  2), 
the  two  As  are  equiangular,  and  the  acute  angles 
which  meet  at  the  extremities  of  B  and  C,  are=to  a  right  angle, 
and  the  lines  B  and  C  make  another  right  angle,  by  construcUon  ; 
therefore,  the  extremities  of  Ay  B,  C,  and  D,  are  in  one  right 
line  (th.  2  b.  1),  and  that  line  is  the  diagonal  of  the  parallelogram 
cb.  Hence,  the  complementary  parallelograms  about  this  parallel- 
ogram are  equal  (th.  28,  b.  I)  ;  but  one  of  these  is  B  long,  and 
and  C  wide,  and  the  other  D  long,  and  A  wide  ;  therefore, 
BXC=AXI).     Q.E.J). 

Cor.     When  B=0  then  A'D=B^,  and  B  is  the  mean  propor- 
tional between  A  and  D. 

THEOREM     22. 

Similar  triangles  are  to  one  another  as  the  squares  of  tlieir  like 
sides. 

Let  ABC,  and  DBF, 
be  two  similar  or  equian- 
gular triangles.  Then  we 
are  to  prove  that 
ABG'.DEF=AB^'.DE^ 
By  the  similarity  of 
the  triangles,  we  have, 
AB 

But,        .        .      AB 

Hence,    ,        .       AB^  .  DE^=^AB'LC  .DE^MF 
But,  by  (th.  30,  b.  1),  AB'LC  is  double  the  area  of  the  A  ABC, 
DE'MF'is  double  of  the  A  DFF. 
Therefore,      ^  ABC  :  A  DFF :  :  AB'LC  :  DE^MF 
(Th.  6,  b.  2).         "  '*     =        AB^\DE\    Q.  E,  D. 


DE=^LG  \MF 
DEz=:AB:  BE 


BOOK    II. 


57 


\4J^ 


THEOREM    23. 

The  perimeters  of  similar  figures  are  to  one  another  as  their  like 
sides  ;  and  their  areas  are  to  one  another  as  the  squares  of  their  like 
sides. 

Let  ABODE,  and  ahcde, 
be  two  similar  figures ;  then 
we  are  to  show  that  EA  is  to 
ea  as  the  sum  of  all  the  sides 
EA-1- AB,  <&c.y  is  to  ea+ab, 
c&c,  and  that  the  area  of  one 
is  to  that  of  the  other,  as  EA^  to  ea',  (yr  AB'  to  ab'. 

As  the  figures  are  exactly  similar  by  bypotbesis,  "whatever  rela- 
tion AB  is  to  EAy  the  same  relation  ah  will  be  to  ea ;  and  if  we 
take 

AB=mEA^ 

'cD^  ""ea  r    ^^®^  ^®  °^^*  ^^^ 

I>E=qEA^ 

Now,  by  (th.  7,  b.  2), 

AE :  ea=:EA-{-mEA,  d;c. 
That  is, 

EA  :  ea=P  :  jp,    P  and  p  representing  the  perimeters  of 
the  figures. 

As  the  two  figures  are  exactly  similar,  whatever  part  the  triangle 
EAB  is  of  one  whole,  the  same  part  the  triangle  eah  is  of  ^ho 
other  whole ;  therefore, 

EAB  :  eab=EABCDE  :  eabcde. 
But  by  (th.  22,  b.  2)  EAB  :  eab=AB^  :  ah" 

Therefore,  by  (th.  6,  b.  2), 

EABCDE  :  eahcde^AB^  :  ah\     Q,  E.  D, 


'  ah=m(ea) 
be  =  n(€a) 
cd=pUa) 

Je=  glea) 


€a-\-meay  <S;c. 


THEOREM    24. 

Two  triangles  which  have  an  angle  in  the  one,  equal  to  an  angle  in 
the  other,  are  to  each  other  as  the  rectangle  of  the  sides  about  the  equal 


58  GEOMETRY. 

Let -4^ C  be  one  triangle,  and  CDE 
the  other,  and  so  placed  that  BC  and 
CD  shall  be  one  and  the  same  line. 
Then  if  the  angle  BCA-=EOD,  ^(7  and  C^  will  be  in  the  same 
line  (converse  of  th.  3,  b.  1).     Draw  the  dotted  line,  AD^  and 
call  the  triangle  ACD=T. 
We  have  now  to  show  that  the 

A  ABC :  A  CDE=.BC'CA  :  GE*CD 
By  (th.  IQ,  b.  2),  A  ABC  :  T=BC  :  CD 
Also,  T\  A  CDE=AC :  CE 

By  multiplying  term  by  term,  and  neglecting  the  common  factor 
in  the  first  couplet,  we  have, 

A  ABC  :  A  CDE=AC'BC  :  CE'CD.    Q.  E.  D. 
Scholium.     When  the  sides  about  the  equal  angles^  are  propor- 
tional, the  two  As  will  be  similar,  and  this  theorem  becomes  essen- 
tially that  of  22  ;  for  in  that  case  we  shall  have, 
BC:  CA=CD:  CE, 
Multiply  the  first  couplet  by  CA,  the  last  couplet  by  CE,  and 
changing  the  means, 

BC'CA  :  CE'CD=CA'  :  CE' 
Comparing  this  proportion  with  the  concluding  one,  we  have, 

A  ABC :  A  CDE=CA'  :  CE' 
Which  is  theorem  22  of  this  book. 


THEOREM    25. 

If  the  vertical  angle  of  a  triangle  he  bisected,  the  bisecting  line  will 
cut  the  base  into  segments,  proportional  to  the  adjacent  sides  of  the 
triangle. 

Let  ABC  be  any  triangle,  and 
bisect  the  vertical  angle,  C,  by  the 
straight  line  CD.  TJien  we  are  to 
show  that 

AD:DB=AC:  CB, 

Produce  AC  to  E,  making 
CE=  CB,  and  join  EB.  The  exterior  angle  A  CB,  of  the  A  CEB, 
is  equal  to  the  two  angles  E,  and  CBE  (th.  16,  b.  1);  but  the 
angle  E=  CBE,  because  CB=  CE;  therefore  the  angle  A  CD,  the 


BOOK    II.  5§ 

half  of  the  angle  ACB^  equals  the  angle  E;  hence,  -DC  and  BE 
are  parallel  (th.  12,  b.  1). 

Now,  as  ABE  is  a  triangle,  and  CD  is  parallel  to  jB (7,  therefore, 
by  (th.  17,  b.  2),  AD  :  DB=AO  :  CE or  C^.     Q.  E.  D. 


THEOREM    26. 

If  from  the  right  angle  of  a  right  angled  triangle,  a  perj^endicular 
he  drawn  to  the  hypotenuse^ 

1 .  The  'perpendicular  divides  the  triangle  into  two  similar  triangles, 
and  each  is  similar  to  the  whole  triangle. 

2.  The  perpendictdar  is  a  mean  proportional  between  the  segments 
of  the  hypotenuse. 

3.  The  segments  of  the  hypotenuse  will  be  in  proportion  to  the 
squares  of  the  adjacent  sides  of  the  triangle. 

4.  The  sum  of  the  squares  of  the  two  sides,  is  equal  to  the  square 
of  the  hypotenuse. 

Let  BAC  be  a  right  angled  triangle, 
right  angled  at  A,  and  draw  AD  perpen- 
dicular to  ^(7.  'PvLtAB=c,  A  C=b,  and 
BC=:a.  Put,  also,  BD=m,  DO=n;  then 
m-\-n=a. 

1.  The  two  As,  ABC,  and  ABD,  have  the  common  angle,  B, 
and  the  right  angle  BAC=BDA;  therefore,  the  third  angle 
C=BAD,  and  the  two  As  are  equiangular,  and  therefore  similar. 
In  the  same  manner  we  prove  the  A  ADC  similar  to  the  A  ABC, 
and  the  two  triangles,  ABD,  ADC,  being  similar  to  the  sjime  A, 
are  similar  to  each  other. 

2.  As  similar  triangles  have  the  sides  about  the  equal  angles 
proportional  (th.  18,  b.  2),  therefore, 

m  :  AD=^AD  :  n;  or,  7n'n=AD^ 

3.  Comparing  the  triangles  ABD,  and  ABC,  the  sides  about  the 
common  angle,  B,  gives 

m  :  c=:c  :  a  (1) 
Comparing  ADC  with.  ABC,  we  have, 

n  :  b=b  :  a  (2) 

From  proportion  (1)  we  have,       am=zc^  (3) 

From        **          (2)      "               aw^i^  (4) 


60  GEOMETRY. 

Divide  equation  (3)  by  (4),  and    —=T^t  which  shows  that  the 

It        0 

ratio  between  n  and  m  is  the  same  as  the  ratio  between  h^  and  c' ;  or, 

n  :  «i=5'  :  c* 
Or,        .        .        .        .         wi  :  n=c^  :  IP 
4.  Add  equations  (3)  and  (4),  and  we  have, 

c^J^h''z=a\n  '\-m)=a\     Q,  E,  D. 
This  last  equation  is  theorem  36,  book  1. 

Scholium.  If  we  take  the  last  equation,  c*-j-J*=o^,  and  trans- 
pose 5^  and  then  separate  the  second  member  into  factors,  we 
shall  have, 

From  this  we  learn  that  in  any  right  angled  triangle,  the  hy- 
potenuse, increased  by  one  side,  multiplied  by  the  hypotenuse 
diminished  by  the  same  side,  is  equal  to  the  square  of  the  other 
side. 


BOOK    III. 


''^\B  H  A  R: 


or  THE  ^ 

UNIVERC 


^  OF 


BOOK     III. 


ON   THB   INVESTIGATION   OP  THE    CIRCLE,  THE   MEASURE   OF  ANGLES, 

AND    OTHER  THEOREMS   IN  WHICH   THE   CIRCLE   IS 

AN   IMPORTANT   ELEMENT. 


DEFINITIONS. 

1 .  A  Curve  Line  is  one  that  is  continually  changing  its  direction. 

2.  A  Circle  is  a  figure  bounded  by  one  uniform  curved  line,  and 
all  straight  lines  drawn  from  a  certain  point  within  it  to  the  curve, 
are  equal ;  and  this  point  is  called  the  center. 

3.  The  entire  curve  is  called  the  circumference  of  the  circle : 
any  portion  of  it  is  called  an  arch,  or  arc  of  the  circle. 

4.  Any  single  straight  Hne  from  the  center  to  the  circumference, 
is  called  the  radius  of  the  circle. 

5.  A  straight  line  drawn  between  any  two  points  on  the  circum 
ference,  is  called  a  chord. 

6.  The  space  on  either  side  of  a  chord, 
inclosed  by  the  chord  and  arc,  is  called  a 
segment  of  a  circle. 

7.  Any  chord  which  passes  through  the 
center,  is  called  a  diameter^  and  such  a  chord 
divides  the  circle  into  two  equal  segments, 
called  semicircles. 

8.  A  straight  line  touching  the  circum- 
ference of  a  circle,  at  any  one  point,  is  called  a  tangent  to  the  circle. 

9.  The  arc,  and  area  between  two  radii,  is  called  the  sector  of 
a  circle. 

Thus  :  the  marginal  figure  represents  a  circle  ;  O  is  the  center, 
CB,  or  CDy  or  CAy  or  any  line  from  C  to  the  circumference,  is  a 
radius.  EGF  is  an  arc ;  EF  is  a  chord ;  the  areas  on  each  side  of 
EF  are  called  segments.  AB  is  a  diameter ;  CBD  is  a  sector;  and 
HD  is  a  tangent. 


f 

a 

\ 

^- 

GEOMETRY. 


THEOREM     1. 

7%e  raditis  perpendicvlar  to  a  chordf  bisects  the  chord,  and  also  the 
arc  of  the  chord. 

Let  ABhe  a>  chord,  C  the  center  of  the 
circle,  and  CD  perpendicular  to  AB  ;  then 
we  are  to  prove  that  AD=BD,  and 
AU=JSB. 

As  C  is  the  center  of  the  circle, 
AC=CB,  and  CD  is  common  to  the  two 
As  A  CD  and  BCD,  and  the  angles  at  D 
being  right  angles,  therefore  the  two  As 
ADC  and  BDC  are  identical,  and  AD=DB,  which  proves  the 
first  part  of  the  theorem. 

Now  as  AD=DB,  and  DU  common  to  the  two  spaces,  ADE 
and  DEB,  and  the  angles  at  D,  right  angles,  if  we  conceive  the 
sector  CBE  turned  over  and  placed  on  GAE,  CE  retHining  its  posi- 
tion, the  point  B  will  fall  on  the  point  A,  because  AD=DB  ;  then 
the  arc  BE  will  fall  on  the  arc  AE;  otherwise,  there  would  be  points 
in  one  or  the  other  arc  unequally  distant  from  the  center,  which 
is  impossible  ;  therefore,  the  arc  AE  =  the  arc  EB.     Q.  E.  D, 

THE  OREM     2. 

Equal  angles,  at  the  center  are  subtended  by  equal  chords, 

(See  figure  to  last  theorem). 
Let  the   angle  AGE=zECB,  then  the  two  isosceles  triangles, 
A  CE,  and  ECB,  are  equal  in  all  respects,  and  AE=EB. 

Q.  E.  D, 
THEOREM     3. 

In  the  same  circle,  or  in  equal  circles,  equal  chords  are  equally 
distant  from  the  center. 

Let  AB  and  EF  be  equal  chords, 
and  C  the  center  of  the  circle.  From 
C,  draw  CO  and  CH  perpendicular  to 
the  respective  chords.  These  perpen- 
diculars will  bisect  the  chords  (th.  1, 
b.  3),  and  we  shall  have  AG=EH, 
W$  ar§  now  to  sh^w  that  C0=:  CH. 


BOOK    III.  63 

In  the  two  As,  A  CG  and  ^Cff,  we  have  ^C=  CA,  A  G=E£r, 
and  the  angle  ^=  the  angle  O,  both  being  right  angles  ;  tliere- 
fore,  the  two  triangles  ACQ,  and  ECU,  are  identical,  and 
CG=CH.     q.E.D. 

We  may  demonstrate  this  theorem  analyticalli/,  and  more  generally ^ 
9^  follows  : 

Let  ^^  represent  the  half  of  any  chord,  and  put  it  equal  to  C. 
r*ut  £[C==F,  and  CE=Ii;  R  representing  the  radius  of  the 
nircle.     Then,  by  (th.  36,  b.  1),  we  have 

C^-{-P^=zR^  (1) 

Also  let  A  G  represent  the  half  of  any  other  chord,  and  put  it 
tqual  to  c;  and  put  its  distance  from  the  center  equal  to  p;  then, 
c2+i?2=^2  (2) 

By  equating  the  first  members  of  (1)  and  (2),  we  have  this 
general  equation :  C''-{-P''=c'-\-p^       (3) 

Now,  if  (7=c,  that  is,  the  chords  equal,  then  P'^=p^^  or  P=p, 
the  perpendiculars  will  be  equal  ;  and  if  P='p,  then  C==c;  that 
is,  chords  equally  distant  from  the  center,  are  equal. 

Equation  (3)  is  true,  under  all  circumstances,  and  if  we  suppose 
C  greater  than  c,  then  P  will  be  less  than  p;  that  is,  the  greater 
the  chord,  the  nearer  it  will  be  to  the  center. 

For  if  0  is  greater  than  c,  let  d  be  their  difference  ; 

Then,         .         .  C=c-\-d,  and  C^=c^-}-2cd+d^ 

And  substitute  this  value  of  C^  in  equation  (3),  and  we  have, 
c'+^cd+d^+P^^c^+p^ 

By  canceling  c^  we  have,  2cd-\-d^-{-  P'^—p^ 

That  is  P^  is  less  than  p^y  because  it  requires  ^cd-^-d^  to  make 
equality  ;  and  if  P^  is  less  than  p^,  P  is  less  than  p;  thai  is,  the 
greater  chord  is  at  a  less  distance  from  the  center. 

Cor.  If  the  chord  C  runs  through  the  center,  then  P,  in  equa- 
tion (3),  equals  0,  and  (7'=c^4-i>^-  ^^*  P^==c^-\-p^,  by  equation 
(2),  or  C^=B?,  or  C=jR,  or  the  semichord  becomes  the  radius, 
a.s  it  manifestly  should,  in  that  case. 

THEOREM    4. 

If  any  line  he  drawn  tangent  to  a  circle,  and  from  the  point  of  con- 
tact a  line  he  drawn  to  the  center  of  the  circhy  the  tangent  and  this 
radiits  will  form  a  right  angle. 

A  tangent  line  can  meet  the  circle  only  at  one  point,  for  if  the 


64 


GEOMETRY. 


line  meets  the  circles  in  two  points,  and  is  still  a  tangent,  it  follows 
that  the  portion  of  the  circumference  between  the  two  points,  is  a 
right  line  ;  but  no  part  of  a  circumference  is  a  right  line,  but  a 
continued  curve  line  ;  and  whenever  a  right  line  meets  a  circle  in 
two  points,  it  must  ctU  the  circle,  and  therefore  cannot  be  a  tangent. 

Now  let  ABO  be  a  tangent  line,  touching 
the  circle  at  the  point  B,  and  draw  the  radius, 
EB,  and  the  line  EC,  and  JSA. 

Now  we  are  to  show  that  EB  is  perpendicular 
to  ABC.  Because  B  is  the  only  point  in  the 
line  ABO  which  touches  the  circle,  any  other 
line,  as  UO,  or  EA,  must  be  greater  than  UB; 
therefore,  £JB  is  the  shortest  line  that  can  be  drawn  from  the  point 
U  to  the  line  AO;  therefore,  UB  is  the  perpendicular  to  AG 
(th.  20,  b.  1).     Q.  K  D. 

THEOREM   5. 

In  the  same  circle,  or  in  equal  circles,  equal  chords  subtend  or  stand 
on  equal  portions  of  the  circumference. 

Conceive  two  equal  circles,  and  two  equal  chords  drawn  within 
them.  Then  conceive  one  circle  taken  up  and  placed  upon  the 
other,  in  such  a  position  that  the  two  equal  chords  will  fall  on,  and 
exactly  coincide  with  each  other;  and  then  the  circles  must  coincide, 
because  they  are  equal ;  and  the  two  segments  of  the  two  circles 
on  each  side  of  the  equal  chords,  must  also  coincide,  or  the  circles 
could  not  coincide  ;  and  magnitudes  which  coincide,  or  exactly  fill 
the  same  space,  are  in  all  respects  equal  (ax.  9).     Therefore 

Q.  E.  D, 
THEOREM    6. 

Thrmngh  three  given  points,  not  in  the  same  straight  line,  one  cir- 
cumference can  be  made  to  pass,  and  but  one. 

Join  AB  and  BO.  If  a  circle  is 
made  to  pass  through  the  two  points 
A  and  B,  the  line  AB  will  be  a  chord 
to  such  a  circle  ;  and  if  a  chord  is 
bisected  by  a  line  at  right  angles,  the 
bisecting  line  will  pass  through  the 
center  of  the  circle  (th.  1,  b.  3)  ; 
therefore,  if  we  bisect  the  line  AB, 


BOOK    III.  65 

and  draw  DF  at  right  angles  from  the  point  of  bisection,  any  circle 
thai  can  pass  through  the  points  A  and  B,  must  have  its  center  some- 
where in  the  line  DF.  And,  by  reasoning  in  the  same  way  (after 
we  draw  £0  at  right  angles  from  the  middle  point  of  JBC),  any 
circle  that  can  pass  through  the  points  B  and  C,  must  have  its 
center  somewhere  in  the  line  FG.  Now,  if  the  two  lines,  DF, 
and  FG,  meet  in  a  common  point,  that  point  will  be  a  center,  from 
whence  a  circle  can  be  drawn  to  pass  through  the  three  points, 
A,  By  and  (7,  and  DF  and  FG  will  always  meet,  unless  they  are 
parallel,  and  if  they  are  parallel,  it  follows  that  AB  and  BC  must 
be  parallel  (definition  1 3),  or  be  in  one  and  the  same  straight  line  ; 
but  this  can  never  be  the  case  while  the  three  given  points,  A^  JB, 
and  (7,  are  not  in  the  same  straight  line ;  therefore  the  two  lines 
will  meet,  and  from  the  point  H,  at  which  they  meet,  a  circle,  and 
only  one  circle,  can  be  drawn,  passing  through  the  three  given 
points.     Q.  F.  D, 


THEOREM     7. 

Jf  tivo  circles  touch  each  other  internally,  or  externally,  the  two 
centers  and  point  of  contact  shall  be  in  one  right  line. 

Let  two  circles  touch  each  other 
internally,  as  represented  at  A,  and 
through  the  point  A,  conceive  AB 
to  be  a  tangent,  at  the  common 
point.  Now,  if  a  line,  perpendic- 
ular to  AB,  be  drawn  from  the 
point  A,  it  must  pass  through  the 
center  of  either  circle  (th.  4,  b.  3);  and  as  there  can  be  but  one 
perpendicular  from  the  same  point,  (th.  20,  b.  1),  therefore.  A,  C, 
and  D,  the  point  of  contact,  and  the  two  centers,  must  be  in  one 
and  the  same  line.     Q.  F.  D. 

Next,  let  the  circles  touch  each  other  externally,  and  from  the 
point  of  contact  conceive  the  common  tangent,  AB,  to  be  drawn. 

Then  a  line,  AC,  perpendicular  to  AB,  will  pass  through  the 
center  of  tlie  external  circle,  (th.  4,  b.  3),  and  a  perpendicular, 
AD,  from  the  same  point,  A,  will  pass  through  the  center  of  the 


66  GEOMETRY. 

other  circle ;  hence,  JBA  C  and  BAD  are  together  equal  to  two 
right  angles;  therefore  (7,  A,  D,  is  one  continued  line  (th.  2, 
b.  1).     Q.U.J), 

Cor.  When  two  circles  touch  each  other  internally,  the  distance 
between  their  centers  is  equal  to  the  diflference  of  their  radii ;  and 
when  they  touch  each  other  externally,  the  distances  of  their 
centers  are  equal  to  the  sum  of  their  radii. 

THEOREM    8. 

An  angle  at  the  circumference  of  any  circle  is  measured  hy  half  the 
arc  on  which  it  stands. 

In  this  work  it  is  taken  as  an  axiom  that  any  angle  standing  at 
the  center  of  a  circle  is  measured  by  the  arc  on  which  it  stands  ; 
and  we  now  proceed  to  show  that  the  angle  at  the  circumference,  is 
half  the  angle  at  the  center. 

Let  A  CB  be  an  angle  at  the  center,  and 
D  an  angle  at  the  circumference,  and  at 
first  suppose  2>  in  a  line  with  A  C.  We  are 
now  to  show  that  the  angle  ACB  is  double  the 
angle  D. 

Join  DB,  and  the  A  DCB  is  an  isosceles 
triangle  ;  for  Ci>=  CB;  and  as  its  exte- 
rior angle,  A  CB,  is  equal  to  the  two  inte- 
rior angles,  i>,  and  CBD,  (th.  11,  b.  1),  and  these  two  angles 
equal  to  each  other ;  therefore,  A  CB  is  double  the  angle  at  D; 
but  ACB  is  measured  by  the  arc  AB;  therefore  the  angle  D  is 
measured  by  half  the  arc  AB. 

Now  let  D  be  not  in  a  line  with  A  C, 
but  at  any  point  on  the  circumference  (ex- 
cept on  AB),  and  join  DC,  and  produce  it 
to^. 

Now  by  the  first  part  of  this  theorem. 

The  angle     .      ECB=^EDB 

Also,    .        .     ECA=<iEDA 

By  subtraction,  ACB=2ADB 

But  ACB  is  measured  by  the  arc  AB ;  therefore  ADD  or  D, 
is  measured  by  one  half  of  the  same  arc.     Q.  E.  D. 


BOOK    III. 


67 


THEOREM    9. 

An  angle  in  a  semicircle^  is  a  right  angle;  an  angle  in  a  segmerdy 
greater  than  a  semicircle,  is  less  than  a  right  angle  ;  and  an  angle  in 
a  segment^  less  than  a  semicircle,  is  greater  than  a  rigid  angle. 

If  the  angle  A  OB  is  in  a  semicircle,  the 
opposite  segment,  ADB,  on  which  it  stands, 
is  also  a  semicircle,  and  the  angle  A  CB  is 
measured  by  half  the  arc  ADB  (th.  8,  b.  2); 
that  is,  half  of  1 80  degrees,  or  90  degrees, 
which  is  the  measure  of  a  right  angle. 

If  the  angle  A  CB  is  in  a  segment  greater 
than  a  semicircle,  then  the  opposite  segment  is  less  than  a  semi- 
circle, and  the  measure  of  the  angle  is  less  than  half  of  1 80  de- 
grees, or  less  than  a  right  angle.  If  the  angle  ACB  is  in  a 
segment  less  than  a  semicircle,  then  the  opposite  segment,  ADB, 
on  which  the  angle  stands,  is  greater  than  a  semicircle,  and  its 
half,  greater  than  90  degrees;  and,  consequently,  the  angle  greater 
than  a  right  angle.     Q.  E.  D. 

Scholium,  Angles  at  the  circumference, 
which  stand  on  the  same  arc  of  a  circle, 
are  equal  to  one  another  ;  for  all  angles,  as 
CAD,  CED,  are  measured  by  half  the 
same  arc,  CD;  and  having  the  same  mea- 
sure, they  must  be  equal. 

Also,  equal  angles  at  the  circumference 
must  stand  on  equal  arcs  ;  for  the  arc,  as 
BC,  and  CD,  being  measures  of  the  angles  BAC,  and  CAD^ 
therefore,  if  the  angles  are  equal,  the  magnitudes,  which  measure 
them,  must  be  equal  also. 

THEOREM     10. 

The  sum  of  two  opposite  angles  of  any  quadrilateral  inscribed  in  a 
circle,  is  equal  to  two  right  angles, 

(See  figure  to  the  last  theorem.) 
Let  ACBD  represent  any  quadrilateral  inscribed  in  a  circle. 
The  angle  A  CB  has  for  its  measure,  half  of  the  arc  ADB,  and 


68 


GEOMETRY. 


the  angle  ADB  has  for  its  measure,  half  of  the  arc  A  CB;  therefore, 
by  addition,  the  sum  of  the  two  oppo>ite  angles  at  C  and  D,  aie 
together  measured  by  half  of  the  whole  circumierence,  or  by  180 
degrees,  or  by  two  right  angles.     Q.  E.  D. 


THEOREM     11. 

An  angle  formed  hy  a  tangent  and  a  chord,  is  measured  hy  one 
half  of  the  intercepted  arc. 

Let  AB  be  a  tangent,  and  AD  a  chord, 
and  ^  the  point  of  contact ;  then  we  are  to 
show  that  the  angle  BAD  is  measured  by  half 
the  arc  AED. 

From  A,  draw  the  radius  AC ;  and  from 
the  center,   C,  draw   CE  perpendicular  to 
AD. 
The  aagle  BAD-\-DA  (7=90°  (th.  4,  b.  3) 

Also,  C+i)^(7=90°  (cor.  4,  th.  11,  b.  I) 

Therefore,  by  subtraction,  BAD — (7=0 
By  transposition,  the  angle        BAD=0. 

But  the  angle  (7,  at  the  center  of  the  circle,  is  measured  by  the 
arc  AE,  the  half  of  AED;  therefore,  the  equal  angle,  BAD,  is 
also  measured  by  the  arc  AE,  the  half  of  AED.     Q.  E.  D. 


THEOREM    12. 

An  angle  formed  by  a  tangent  and  a  chord,  is  equal  to  an  an^le  in 
the  opposite  segment  of  the  circle. 

Let  AB  be  a  tangent,  and  AD  a  chord^ 
and  from  the  point  of  contact.  A,  draw 
any  angles,  as  A  CD,  and  AED,  in  the  seg- 
ments. Then  we  are  to  shoio  that  the  angle 
BAD=ACD,  and  GAD=AED. 

By  the  last  theorem,  the  angle  BAD  is 
measured  by  half  the  arc  AED;  and  as  the 
angle  ACD  (th.  8,  b.  3)  is  measured  by 
half  of  the  same  arc,  therefore  the  angle  BAD=ACD 


BOOK     III.  69 

Again,  as  AEDG  is  a  quadrilateral,  inscribed  in  a  circle,  the 
sum  of  the  opposite  angles, 

AGD-\-AED=^2  right  angles,     (th.  10,  b.  3) 
Also,  the  angles  BAD-\-DAG=^  right  angles,     (th.    1,  b.  1) 
By  subtraction  (and  observing  that  BAD  has  just  been  proved 
equal  to  A  CD),  vre  have, 

AI!J)^J)AG=0 
Or,        •        •        •         AED=DA  G,  by  transposition. 

Q.  E.  D, 


THEOREM     13. 

Parallel  chords^  or  a  tangent  and  a  parallel  chord,  intercept  equal 
a/rcs  on  the  circumference. 

Let  AB  and  GD  be  two  parallel  chords, 
and  draw  the  diagonal,  AD;  and  because 
AB  and  GD  are  parallel,  the  angle  DAB 
=  the  angle  ADG  (th.  5,  b.  1);  but 
the  angle  DAB  has  for  its  measure,  half 
of  the  arc  BD;  and  the  angle  ADG  has 
for  its  measure,  half  of  the  arc  AG  (th.  8,  b.  3);  and  because 
the  angles  are  equal,  the  arcs  are  equal ;  that  is,  the  arc  BD—  the 
arc  ^(7.     q.E.D, 

Next,  let  EF  be  a  tangent,  parallel  to  a  chord,  GD,  and  from 
the  point  of  contact,  G,  draw  GD, 

By  reason  of  the  parallels,  the  angle  GDG  =  the  angle  DGF, 
But  the  angle  CDG  has  for  its  measure,  half  of  the  arc  GO  (th. 
9,  b.  3);  and  the  angle  DGF  has  for  its  measure,  half  of  the 
arc  GD  (th.  11,  b.  3);  therefore,  these  equal  measures  of  equals 
must  be  equal ;  that  is,  the  arc  GG  =  the  arc  GD,     Q,  E.  D, 


THEOREM     II. 

When  two  chords  intersect  each  other  within  a  circle,  the  angle  thus 
formed  is  measured  by  half  the  sum  of  the  two  intercepted  arcs. 


70 


GEOMETRY 


Let -45  and  CD  intersect  each  other 

within  the  circle  forming  the  two  angles, 
£,  and  E^y  with  their  opposite  vertical 
and  equal  angles. 

Then  we  are  to  show,  that  the  angle  E  is 
measured  by  the  half  sum  of  the  arcs 
AC-j-BD;  and  the  angle  E^  is  measured 
by  the  half  sum  of  the  arcs  AD-j-CB. 

First,  draw  AF  parallel  to  CD;  then, 
by  reason  of  the  parallels,  the  angle  BAF=E.  But  the  angle 
BAF  is  measured  by  half  of  the  arc  FDB;  that  is,  half  of  the 
arc  BD,  plus  half  of  the  arc  A  C;  because  FD=A  C  (th.  1 3,b.  3). 

Now,  as  the  sum  of  the  angles,  E-\-F^y  make  two  right  angles, 
that  sum  is  measured  by  half  the  whole  circumference. 

But  the  angle  Ey  alone,  as  we  have  just  determined,  is  mea- 
sured by  half  the  sum  of  the  arcs  BD-\-A  C;  therefore,  the  other 
angle,  -fi'S  is  measured  by  half  of  the  other  parts  of  the  circum- 
ference, AD-^  CB,     Q.  E.  D. 


THEOREM     15 


When  two  chords  intersect,  or  meet  each  other  without  a  circle,  the 
angle  fhtcs  formed  is  measured  hy  half  the  difference  of  the  intercejpted 
arcs. 

Draw  AF  parallel  to  CD;  then,  by 
reason  of  the  parallels,  the  angle  E,  made 
by  the  intersection  of  the  two  chords,  is 
equal  to  the  angle  BAF.  Biit  the  angle 
BAF  is  measured  by  half  the  arc  BF; 
that  is,  by  half  the  diflference  between 
the  arcs  BD  and  A  C.     Q.  E.  D. 

N.  B.  Prolonged  chords,  to  meet 
without  the  circle,  as  ED,  and  EB,  are 
called  secants.  They  are  geometrical, 
and  not  trigonometrical  secants. 


BOOK   in 


71 


THEOREM     16. 

The  angle  formed  hy  a  secant  and  a  tangent,  is  measured  ly  half  the 
difference  of  the  intercepted  arcs. 

Let  CB  be  a  secant,  and  CD  a  tangent. 
We  are  now  to  show  that  the  angle  formed  at 
C,  is  measured  hy  half  of  the  difference  of 
the  arcs  BD  and  DA. 

From  A,  draw  AE  parallel  to  CD;  then 
the  angle  BAE=C.  But  the  angle  BAE 
is  measured  by  half  of  the  arc  BE  (th.  8, 
b.  3);  that  is,  by  half  of  the  diflference  be- 
tween the  arcs  BD  and  AD;  for  the  arc 
AD=DE,  and  BD—DE=BE;  therefore  the  equal  angle,  (7,  is 
measured  by  half  the  arc  BE,     Q.  E.  D, 


THEOREM     17. 

When  two  chords  intersect  each  other  in  a  circle,  the  rectangle  of  the 
segments  of  the  one,  will  be  egtml  to  the  rectangle  of  the  segments  of  the 
other. 

Let  AB  and  CD  be  two  chords  intersect- 
ing each  other  in  E,  Then  we  are  to  show 
that  the  rectangle  A  EX  EB=  CEX  ED. 

Join  AD  and  CB,  forming  the  two  tri- 
angles A  ED  and  CEB,  which  are  equi- 
angular, and  therefore  similar ;  for  the 
angles  B  and  D  are  equal,  because  they  are 
both  measured  by  half  the  arc  A  C.  Also  the  angles  A  and  C  are 
equal,  because  each  is  measured  by  half  the  same  arc,  DB;  and 
the  angle  AED—CEB,  because  they  are  vertical  angles;  hence, 
the  triangles,  AED  and  CEB  are  equiangular.  But  equiangular 
triangles  have  their  sides,  about  the  equal  angles,  proportional 
(th.  18,  b.  2);  therefore,  AE  and  ED,  about  the  angle  E,  are  pro- 
portional to  CE  and  EB,  about  the  same  angle. 

That  is,         .         .       AE  :  ED  :  :  CE  :  EB 

Or  (th.  21,  b.  2),  AEXEB=^  EDXEC.     Q.  E.  D. 


72 


GEOMETRY. 


Scholium.  When  one  chord  is  a  diameter,  and  the  other  at  right 
angles  to  it,  the  rectangle  of  the  segments  of  the  diameter  is  eqvxd  to  the 
square  of  half  the  other  chord;  cyr  half  of  the  bisected  chcyrd  is  a  mean 
proportional  between  the  segments  of  the  diameter. 


Y    n      >f 


For  ADXDB==FDXDE.  But  if  AB 
passes  through  the  center,  C,  at  right  an- 
gles to  FE,  then  FD=DE  (th.  1,  b.  3), 
and  in  the  place  of  FD,  write  its  equal,  DF, 
in  the  last  equation,  and  we  have, 
ADXDB=DF' 

Or,        .     ADxDEi'.DE'.DB 

Put,  DE=x,  CD=y,  and  CE=^R,  the  radius  of  the  circle. 

Then       ^i>=R— y,  and  DB=Ji-\-g.       With    this    notation, 
ADXDB, 

Becomes,     .        .  (JR — g)(Ii-\-g)=x^ 

Or,      ...         .       Ii'—y^=a^ 

Or,      .        .         .         .         .      B^=x^+7/^ 

That  is,  the  square  of  the  hypotenuse  of  the  right  angled  triangle, 
DOE,  is  equal  to  the  sum  of  the  squares  of  the  other  two  sides.    , 


THEOREM     18. 

If  from  any  point  withmii  a  circle,  any  number  of  secants  be  dravm, 
the  rectangle  formed  hy  any  one  secant  and  its  external  segment,  will 
be  equal  to  the  rectangle  of  any  other  secant,  and  its  external  segment. 

Let  AB,  A  C,  AD,  <kc.,  be  secants,  and 
AE,  AF,  AGs  &c.,   their   external   seg- 
ments.    Then  we  are  to  show  that 
ABXAE=ACXAF 

And,      AB  X  AE=AD  XAG,  &c. 

Join  BF  and  EO;  then  the  two  As, 
AFB  and  AEO  are  equiangular  ;  for  the 
angle  B=  C,  as  each  of  them  is  measured 
by  half  the  same  arc,  EF;  and  the  angle 
BA  C  is  common  to  the  two  triangles ; 
therefore,  the  third  angles  are  equal  (th.  11,  cor.  2,  b.  l). 


BOOK    HI.  73 

Therefore  (th.  18,  b.  2),    AB  :  AF :  :  AC ',  AE 
Hence,         .        .        .       ABXAE^ACXAF 

In  the  same  manner  we  may  prove  that 

ABXAE=AQKAD 
And,  ....       ACXAF=AQXAD 

Q.  E.  D. 

Scholium  1.  If  we  conceive  AD  to  revolve  outward,  on  Ay  as  a 
fixed  point,  0-  and  D  will  come  nearer  together,  and  will  be 
exactly  together  in  the  tangent  All* 

But  however  far  or  near  G  may  be  to  D,  we  always  have, 

ABXAE=ADXAG 
And,  when  both  AD  and  A  G  become  AH^  we  shall  have, 

ABy.AE==AB} 

Scholium  2.  If  AH  and  AP  be  tangents  to  the  same  circle, 
from  the  same  point  on  each  side  of  A,  they  will  be  equal  to  each 
other ; 

For,        .         BAXAE=:AP^ 

Also,       .        BAXAE=AW 

Hence  (ax.  1 ),      {AF')=(AIP),  or  AP=Aff, 

This  property  will  enable  us  to  compute  the  diameter  of  the  earth,  when- 
ever we  know  the  visible  distance  of  its  regular  surface,  as  seen  from  any 
known  hight  above  the  surface. 

For  example,  suppose  FC  to  be  the  diameter  of  the  earth,  AFj  the  hight 
of  a  mountain,  and  AH  the  distance  on  sea  to  the  visible  horizon.  If  AF 
and  AH  were  both  known,  FC  could  be  computed  therefrom.  For,  let  FC 
r=x,  AF=:h,  and  AH=d. 

Then,        .        .        .     (A4^)*=rf2,  or  ar=?— A 

h 

On  this  principle,  rough  estimates  of  the  diameter  of  the  earth  have  been 

made  ;  and  on  this  principle  the  dip  of  the  horizon  has  been  computed. 


THEOREM     19. 

^  a  circle  he  described  about  a  triangle,  the  rectangle  of  two  sides 
is  equal  to  the  rectangle  of  the  perpendicular  let  fall  on  to  the  third  side, 
and  the  diameter  of  the  circumscribing  circle. 


t4  GEOMETRY. 

Let  ABC  be  the  triangle,  AC  and  CJ5, 
the  sides,  CD  the  perpendicular  on  the  base, 
and  CB  the  diameter  of  the  circle.  Then  we 
are  to  shmo  that 

ACXCB^CEXCJD, 

The  two  As,  A  CD  and  CEB,  are  equian- 
gular, because  -4=-£',both  measured  by  the 
half  of  the  arc  CB.  Also,  AD  (7  is  a  right  angle,  equal  to  CBE^ 
an  angle  in  a  semicircle,  and  therefore  a  right  angle ;  hence,  the 
third  angle,  ACD=BCE  (th.  11,  cor.  1,  b.  1).  Therefore 
(th.  18,  b.  2), 

AC  :  CD  :  :  EC  :  CB 

Hence,        .        .   ACxCB=CEXCD.  Q.J^.D. 

Scholium.  The  continued  proditct  of  three  sides  of  a  triangle,  w 
equal  to  the  double  area  of  the  triangle  into  the  diameter  of  its  drcum- 
scribing  circle. 

Multiply  both  members  of  the  last  equation  by  AB,  and  we  haye, 

ACX  CBXAB=CEX{ABxCD) 
But  CE  is  the  diameter  of  the  circle,  and  {ABx  CD)  =  twice 
the  area  of  the  triangle  ; 

Therefore,       ACxCBXAB=^  diametei-  X 2 As. 


THEOREM     20. 

The  square  of  a  line  bisecting  any  angle  of  a  triangle,  together  vnih 
the  rectangle  of  the  segments  it  makes  iviih  the  opposite  side,  are  equal 
to  the  rectangle  of  the  two  sides,  including  the  bisected  angle. 

Let  ABC  be  the  triangle,  CD  the  line  bi- 
secting the  angle  C.  Then  we  are  to  show  thai 
CD'-{-ADxDB=ACX  CB, 
The  two  As,  ACE  and  CDB,  are  equi- 
angular, because  the  angles  E  and  B  are 
equal,  both  being  in  the  same  segment,  and 
the  J  ACE=^BCD,  by  hypothesis.  There- 
fore, (th.  18,  b.  2), 

AC  :  CE::  CD:  CB 


BOOK    III.  75 

But  it  is  obvious  that  CE=^CD-\-DEy  and  by   substituting 
this  value  of  CEt  in  the  proportion,  we  have, 

AC:(OD-{-DE)::CD:  CB 
By  multiplying  extremes  and  means, 

CJ)'-\-I>EX  CD=ACX  CB 
But  DEX  CD=AI>XDB,  by  (th.  17,  b.  3),  which,  being  sub- 
tituted,  we  have, 

CJD'+AD  X  DB=A  OX  CB.     Q.  E,  J), 


THEOREM     21. 

The  rectangle  of  the  two  diagonals  of  any  quadrilateral  inscribed  in  a 
circle,  is  equal  to  the  sum  of  the  two  rectangles  of  the  opposite  sides. 

Let  ABCD  be  a  quadrilateral  in  a  circle  ; 
then  ice  are  to  show  that 

ACX  BD=:ABX  DC-\-ADxBC, 

From  C,  let  CE  be  drawn  so  that  the 
angle  DCE  shall  be  equal  to  angle  ACB; 
and  as  the  angle  BAC  is  equal  to  the 
angle  CDE,  both  being  in  the  same  seg- 
ment, therefore,  the  two  triangles,  DEC  and  ABC  are  equiangular, 
and  we  have  (th.  18,  b.  2), 

AB\AC\\DE\DC        (1) 

The  two  As,  ADC  and  BEC  are  equiangular;  for  the  angle 
DAC=EBC,  both  being  in  the  same  segment,  are  measured  by 
half  the  same  arc,  i>(7;  and  the  a,ng\e  DCA=zECB;  for  DCE 
=BCA;  and  to  each  of  these  add  the  angle  EGA,  and  DCA 
^ECB;  therefore  (th.  18,  b.  2), 

AD:  AC::  BE:  BC        (2) 
By  multiplying  the  extremes  and  means  in  these  two  proportions, 
and  adding  the  equations  together,  we  have, 

(ABXDC)-{'{ADxBC)=={DE-{-BE)XAO 
But,         .        .        .      DE-\-BEz=BD;  therefore, 

{ABXDC)-\'{ADXBC)==BDXAC     Q,  E,  D. 


c 


76  GEOMETRY. 

Scholium.    When  two  of  the  adjacent  sides  of  the  quadrilateral 
are  equal,  as  AB=BC,  then  the  resulting  equation  is, 
(ABXD0)+(ABXAI>)=BDXAC 
Or,        .        .  ABX(DC-{-AJ))=BDxAC 
Or,        .        .        .       AB',AC:'.BD:{CD-\-AD) 
That  is,  as  one  of  the  equal  sides  of  the  quadnlateral,  is  to  the 
adjoining  diagonal,  so  is  the  transverse  diagonal  to  tfie  sum  of  the  two 
unequal  sides. 

THEOREM    22. 

If  two  chords  intersect  each  other  in  a  circle,  at  right  angles,  the  sum 
of  the  squares  of  the  four  segments  thu^  formed,  is  equal  to  the  square 
of  the  diameter  of  the  circle. 

Let  AB  and  CD  be  two  chords,  intersect- 
ing each  other  at  right  angles.  Draw  BF 
parallel  to  ED,  and  join  BF  and  AF.  Now 
we  are  to  show  that 

AE'-\-EB^-\-E(P-\-ED^=AF', 

As  BF  is  parallel  to  ED,  ABF  is  a  right 
angle,   and    therefore    AF  is    a    diameter 
(th.  9,  b.  3).     Also,  because  BF  is  parallel  to   CD,   CB=DF 
(th.  13,  b.  3). 

Because  CEB  is  a  right  angle,    .     CE''-\-EB''=z  CE'=DF' 

Because  AED  is  a  right  angle,     .    AE^-\-EB''=AD'' 
Adding  these  two  equations,  we  have, 

CF?4rEB'''\-AE''^ED''=DF^-\-AD^ 
But,  as  AF  is  a  diameter,  and -42>^  a  right  angle  (th.  9,  b.  3), 
Therefore         .    DF^-\-AD''^AF^ 
Hence,     .        .    CE^^-EB^-^AE^-^ED^^AF^         Q.  E.  D. 

Scholium.  If  two  chords  intersect  each  other  at  riofht  angles, 
111  a  circle,  and  their  opposite  extremities  be  joined,  the  two  chords 
thus  formed  may  make  two  sides  of  a  right  angled  tripngle,  of 
which  the  diameter  of  the  circle  is  the  hypotenuse. 

For  AD  is  one  of  these  chords,  and  CB  is  the  other ;  and  we 
have  shown  that  CB=DF;  and  AD  and  DF  are  two  sides  of  a 


BOOK    III.  -n 

right  angled  triangle,  of  which  AF  is  the  hypotenuse ;  therefore, 
AD  and  OB  may  be  considered  the  two  sides  of  a  right  angle, 
and  AF  \\&  hypotenuse. 

THEOREM     23. 

If  two  secants  intersect  each  other  at  rigid  angles,  the  sinn  of  their 
squares,  increased  by  the  sum  of  the  squares  of  the  two  parts  without 
the  circle,  vnll  he  equal  to  the  square  of  the  diameter  of  the  circle. 

Let  AE  and  ED  be  two  secants  intersect- 
ing at  right  angles  at  the  point  E.  From  B^ 
draw  BF  parallel  to  CD,  and  join  AF  and 
AD.     ybw  we  are  to  show  that 

EA'-^ED'-{-EB'+EC^=AF^ 

Because  BF  is  parallel  to  CD,  ABF  is  a 
right    angle,    and   consequently   ^^   is    a 
diameter,  and  BC—DF;  and  because  AF  is  a  diameter,  ADFis 
a  right  angle.     As  AED  is  a  right  angle, 
AE^^ED^=AD^ 

Also,        .        .         EB^'\-EC^=BC^=DF^ 


J3  _E 


\    Jl> 


By  addition,    AE'^-\-ED''+EB^+EC^=^AD'-^DF^^AF\ 

Q.  E,  D. 


78 


GEOMETRT. 


BOOK     IV 


PROBLEMS. 

In  this  section,  we  shall,  in  most  instances,  merely  show  the 
construction  of  the  problem,  and  refer  to  the  theorem  or  theorems 
that  the  student  may  use,  to  prove  that  the  object  is  attained  by  the 
construction. 

In  obscure  and  difficult  problems,  however,  we  shall  go  through 
the  demonstration  as  though  it  were  a  theorem. 


PROBLEM    I  . 

To  bisect  a  given  finite  straight  line. 

Let  AB  be  the  given  line,  and  from  its 
extremities,  A  and  jB,  with  any  radius 
greater  than  the  half  of  -^^  (Post.  3),  de- 
scribe arcs,  cutting  each  other  in  n  and  m. 
Join  n  and  m;  and  C,  where  it  cuts  AB 
wilj  be  the  middle  of  the  line  required. 

Proof,  (th.  15,  b,  1,  cor.  1  ). 


PROBLEM     2. 

To  bisect  a  given  angle. 

Let  ABC  be  the  given  angle.  With  any 
radius,  from  the  center  B,  describe  the  arc 
AC.  From  A  and  G,  as  centers,  with  a 
radius  greater  than  the  half  of  AC,  de- 
scribe arcs,  intersecting  in  n;  and  join  Bn, 
it  will  bisect  the  given  angle. 

Proof,  (th.  19,  b.  1). 


BOOK    IV. 


79 


PROBLEM    3. 

From  a  given  poirUf  in  a  given  line,  to  draw  a  perpendicular  to  thai 
line. 

Let  AB  be  the  given  line,  and  C 
the  given  point.  Take  n  and  m  equal 
distances  on  opposite  sides  of  C;  and 
from  the  points  m  and  n,  as  centers, 
with  any  radius  greater  than  nO  or 
or  mCj  describe  arcs  cutting  each  other 
in  S.  Join  SC,  and  it  will  be  the  per- 
pendicular required.     Proof,  (th.  15,  b.  1,  cor.     ) 

The  following  is  another  method,  which 
is  preferable,  when  the  given  point,  (7,  is  at 
or  near  the  end  of  the  line. 

Take  any  point,  0,  which  is  manifestly 
one  side  of  the  perpendicular,  and  join  OC; 
and  with  0(7,  as  a  radius,  describe  an  arc, 
cutting  AB  in  m  and  C.  Join  m  0,  and  produce  it  to  meet  the 
arc,  again,  in  n;  mn  is  then  a  diameter  to  the  circle.  Join  Cn^  and 
it  will  be  the  perpendicular  required.     Proof,  (th.  9,  b.  3). 


PROBLEM    4. 

From  a  given  point  vnthotU  a  line,  to  draw  a  perpendicular  to  thai 
Ime. 

Let  AB  be  the  given  line,  and  0  the 
given  point.  From  0,  draw  any  oblique 
line,  as  Cn,  Find  the  middle  point  of 
Cn  by  (problem  1),  and  from  that  point, 
as  a  center,  describe  a  semicircle,  having 
Cn  as  a  diameter.  From  the  point  m, 
where  this  semicircle  cuts  AB,  draw  Cm, 
and  it  will  be  the  perpendicular  required. 


Proof,  (th.  9,  b.  3). 


eo 


GEOMETRY. 


PROBLEM    5. 

At  a  ^ven  point  in  a  line,  to  make  an  angle  equal  to  anMer  given 
angle. 

Let  A  be  the  given  point  in  the  line  AB, 
and  DOE  the  given  angle. 

From   (7  as  a  center,  with  any  radius, 
CE,  draw  the  arc  ED. 

From  ^,  as  a    center,  with   the  radius 
AF=  CE,  describe  an  indefinite  arc ;  and 
from  F,  as  a  center,  with  FG  as  a  radius, 
equal  to  ED,  describe  an  arc,  cutting  the  other  arc  in  G,  and  join 
AG;  GAF  will  be  the  angle  required.     Proof,  (th.  5,  b.  3). 


PROBLEM    6. 

From  a  given  point,  to  draw  a  line  parallel  to  a  given  line. 

Let  A  be  the  given  point,  and  CB  the 
given  line.  Draw  AB,  making  an  angle, 
ABC;  and  from  the  given  point.  A,  in  the 
line  AB,  draw  the  angle  BAD=ABC,  by 
the  last  problem. 

AD  and  CB  make  the  same  angle  with  AB;  they  are,  therefore, 
parallel.     (Definition  of  parallel  lines). 


PROBLEM    7. 

To  divide  a  given  line  into  any  number  of  equal  parts. 
LeiAB  represent  the  given  line,  and 
let  it  be  required  to  divide  it  into  any 
number  of  equal  parts,  say  five.  From 
one  end  of  the  line  A,  draw  AD,  inde- 
finite in  both  length  and  position.  Take 
any  convenient  distance  in  the  dividers, 
as  Aa,  and  set  it  oflf  on  the  line  AD; 
thus  making  the  parts  Aa,  ab,  be,  <fec.,  equal.  Through  the  last 
point,  e,  draw  EB,  and  through  the  points  a,  b,  c,  and  d,  draw 
parallels  to  eB  (problem  6.);  these  parallels  will  divide  the  line  as 
required      Proof  (th.  17,  b.  2). 


BOOK    IV. 


81 


PROBLEM    8. 

To  find  a  third  proportional  to  two  given  lines 

Let  AB  and  AC  he  any  two  lines.     Place     A 

them  at  any  angle,  and  join  CB.  On  the 
greater  line,  AB,  take  AD=A  G,  and  through 
Dy  draw  DE  parallel  to  BC;  AE  is  the  third 
proportional  required. 

ProoC  (th.  17,  b.  2). 


B 


PROBLEM    9. 

To  find  a  fourth  proportional  to  three  given  lines 

Let  ABy  AC,  AD,  represent  the 
three  given  lines.  Place  the  first  two 
together,  at  a  point  forming  any  angle, 
as  BACy  and  join  BC.  On  AB  place 
ADy  and  from  the  point  i>,  draw 
(problem  6)  DE  parallel  to  BC;  AE 
will  be  the  fourth  proportional  required. 

Proof,  (th.  17,  b.  2). 


PROBLEM     10. 

To  find  the  middle,  or  mean  proportional,  between  two  given  lines. 

Place  AB  and  BC  in  one  right  line, 
and,  on  AC,  as  a  diameter,  describe  a 
semicircle  (postulate  3),  and  from  the 
point  B,  draw  BJ)  at  right  angles  to  AC 
(problem  3);  BD  is  the  mean  propor- 
tional required. 

Proof,  (scholium  to  th.  17,  b.  3). 
6 


82 


GEOMETRY 


PROBLEM     11. 

To  find  the  center  of  a  given  circle. 

Draw  any  two  chords  in  the  given  circle, 
as  AB  and  CD;  and  from  the  middle  point, 
w,  of  ABf  draw  a  perpendicular  to  AB; 
and  from  the  middle  point,  m,  draw  a  per- 
pendicular to  CD;  and  where  these  two 
perpendiculars  intersect  will  be  the  center 
of  the  circle.     Proof,  (th.  1,  b.  3). 


PROBLEM     12. 

To  draw  a  tangent  to  a  given  circle,  from  a  given  point,  either  in 
<yr  imthout  the  circumference  of  the  circle. 

When  the  given  point  is  in  the  circum- 
ference, as  A,  draw  AG  the  radius,  and 
from  the  point  A,  draw  AB  perpendicular 
to  A  C;  AB  is  the  tangent  required. 

Proof,  (th,  4,  b.  3). 


When  A  is  without  the  circle,  draw 
^  C  to  the  center  of  the  circle  ;  and  on 
AO,  as  a  diameter,  describe  a  semi- 
circle ;  and  from  the  point  B,  where 
this  semicircle  intersects  the  given 
circle,  draw  AB,  and  it  will  be  tangent 
to  the  circle. 

Proof,  (th.  9,  b.  3),  and  (th.  4,  b.  3). 


PROBLEM     18. 

On  a  given  line,  to  describe  a  segment  of  a  circle,  thai  shall  contain 
an  angle  equal  to  a  given  angle. 


BOOK    IV. 


89 


Let  AB  he  the  given  line,  and  O 
the  given  angle.  At  the  ends  of  the 
given  line,  make  angles  DAB,  DBA, 
each  equal  to  the  given  angle,  C. 
Then  draw  A£J,  jB^,  perpendiculars  to 
AD^  JBD;  and  with  the  center,  JEJ,  and 
radius,  UA  or  £B,  describe  a  circle  ; 
then  AFB  will  be  the  segment  required,  as  any  angle  -^,  made  in 
it,  will  be  equal  to  the  given  angle,  C. 

Proof,  (th  11.  b.  3),  and  (th.  8,  b.  3). 


PROBLEM     14. 

To  cut  a  segment  fT(ym  any  given  circle^  that  shall  contain  a  given 
angle. 

Let  C  be  the  given  angle.  Take 
any  point,  as  ^,  in  the  circumference, 
and  from  that  point  draw  the  tangent 
AB;  and  from  the  point  A,  in  the  line 
ABy  make  the  angle  BAD=0  (pro- 
blem 5),  and  A.ED  is  the  segment 
required. 

Proof,  (th.  11,  b.  3),  and  (th.  8,  b.  3) 


PR^OBLEM     15. 

To  construct  an  equilateral  triangle  on  a  given  finite  straight  line. 

Let  AB  be  the  given  line,  and  from  one 
extremity.  A,  as  a  center,  with  a  radius 
equal  to  AB,  describe  an  arc.  At  the  other 
extremity,  B,  with  the  same  radius,  describe 
another  arc.  From  C,  where  these  two 
arcs  intersect,  draw  CA  and  CB;  -4^(7  will 
be  the  triangle  required. 

Tlie  construction  is  a  sufficient  demonstration.     Or,  (ax.  1). 


z 


84 


GEOMETRY. 


D 


PROBLEM     16. 

To  construct  a  triangle,  having  its  three  sides  equal  to  three  given 
lines,  any  two  of  which  shall  be  greater  than  the  third. 

Let  AB,   CD,  and  EF  represent  the  three      E F 

lines.  Take  any  one  of  them,  as  ABy  to  be  one 
side  of  the  triangle.  From  -4,  as  a  center,  with 
a  radius  equal  to  CDy  describe  an  arc;  and 
from  B,  as  a  center,  with  a  radius  equal  to  EF, 
describe  another  arc,  cutting  the  former  in  n. 
Join  An  and  Bn,  and  AnB  will  be  the  A 
required.    Proof,  (ax.  1). 


PROBLEM     17. 

To  describe  a  square  on  a  given  line. 

Let  AB  be  the  given  line,  and  from  the  extre- 
mities, A  and  By  draw  AQ  and  BD  perpendicular 
ioAB.     (Problems.) 

From  -4,  as  a  center,  with  AB  as  radius,  strike 
an  arc  across  the  perpendicular  at  C;  and  from  C, 
draw  CD  parallel  to  AB;  A  CDB  is  the  square 
required.     Proof,  (th.  21,  b.  1.) 


B 


PROBLEM     18. 

To  construct  a  rectangle^  or  a  parallelogram,  whose  adjacent  sides 
are  equal  to  two  given  lines. 

Let  AB  and  AC  he  the  two  given  lines.      A C 

From  the  extremities  of  one  line,  draw  per-      A B 

pendiculars  to  that  line,  as  in  the  last  problem  ;  and  from  these 
perpendiculars,  cue  oflf  portions  equal  to  the  other  line  ;  and  by  a 
parallel,  complete  the  figure. 

When  the  figure  is  to  be  a  parallelogram,  with  oblique  angles, 
describe  the  angles  by  problem  5.     Proof,  (th.  21,  b.  1). 


BOOK    IV. 


89 


PROBLEM     19. 

To  describe  a  rectangle  that  shall  he  eqiial  to  a  given  square^  and 
have  a  side  equal  to  a  given  line* 

Let  AB  be  a  side  of  the  given  square,  and      C D 

CD  one  side  of  the  required  rectangle.  A B 

Find   the  third  proportional,  EF,  to  CD      E F 

and  AB  (problem  8).     Then  we  shall  have, 
CD\AB\\AB\EF 

Construct  a  rectangle  with  the  two  given  lines,  CD  and  EF 
(problem  18),  and  it  will  be  equal  to  the  given  square,  (th.  3,  b,  2). 

PROBLEM    20. 

To  construct  a  square  that  shall  he  equal  t9  the  difference  of  tvio 
given  squares. 

Let  A  represent  a  side  of  the  greater  of  two  given  squares,  and 
B  a  side  of  the  lesser  square. 

On  ^,  as  a  diameter,  describe  a  semi- 
circle, and  from  one  extremity,  j?,  as  a  cen- 
ter, with  a  radius  equal  to  B^  describe  an 
arc, »,  and,  from  the  point  where  it  cuts  the 
circumference,  draw  mn  and  np;  np  is  the 
side  of  a  square,  which,  when  constructed, 
(problem  17),  will  be  equal  to  the  difference 
of  the  two  given  squares.     Proof,  (th.  9,  b.  3,  and  36,  b.  1.) 

PROBLEM    21. 

To  construct  a  square,  that  shall  he  to  a  given  square,  as  a  line,  M, 
to  a  line,  N. 

Place  M  and  iV  in  a  line,  and  on  the  sum  describe  a  semicircle. 
From  the  point  where  they  join,  draw  a  perpendicular  to  meet  the 
circumference  in  A.  Join  Am  and 
An,  and  produce  them  indefinitely. 
On  Am  or  An,  produced,  take  AB= 
to  the  side  of  the  given  square  ;  and 
from  B,  draw  BG  parallel  to  mn; 
-4  C  is  a  side  of  the  required  square. 


86 


GEOMETRY. 


For,  Am^  :  An^  :  :  AB^ 

Also,         Am^  :An^::M 
Therefore,  AJB'iAC^:  :M 


AC' 


(th.  17,  b.  «.) 
(scholium  to  th.  26,  b.  2.) 
(th.  6,  b.  2.)    Q,  E.  D, 


T\' 


PROBLEM    22. 

To  cut  a  line  into  extreme  and  mean  ratio;  thai  is,  so  that  the  whole 
shall  he  to  the  greater  part,  as  that  greater  is  to  the  less. 

Let  AB  be  the  line,  and  from  one  extre- 
mity, B,  draw  BC  dX  right  angles,  and  equal 
to  half  AB. 

From  (7,  as  a  center,  and  radius  CB,  de- 
scribe a  circle.  Join  A  C  and  produce  it  to 
F.  From  -4,  as  a  cepter,  and  AD  radius, 
describe  the  arc  DE;  this  arc  will  divide  the 
line  AB,  as  required. 

We  are  now  to  shoio  that 

AB  :AE::AE:EB 

By  (scholium  to  th.  18,  b.  3),  we  have, 

AFXAD=AB' 
Or,        .        .         AF:AB::AB:AD 
Then,  by  (th.  8,  b.  2),  we  may  have, 

(AF—AB)  :AB::  (AB—AD)  :  AD 
As        .         .  (7^=i^/?=^i>i^;  therefore,  ^^--Z»/' 

Hence, .        .        .  AF—AB=AF^DF=AD=AE 
Therefore,     .        .  AE :  AB  :  :  EB  :  AE 
By  taking  the  extremes  for  the  means,  we  have, 

AB:AE::AE:EB  Q.  E.  D. 


PROBLEM    2S. 

To  describe  an  isosceles  triangle,  having  its  two  equal  angles  double 
of  the  third  angle,  and  the  equal  sides  of  any  given  length. 


BOOK    IV.  87 

Let  AB  be  one  of  the  equal  sides  of  the 
required  triangle ;  and  from  the  point  A, 
with  AB  radius,  strike  an  arc,  BD. 

Divide  the  line  AB  into  extreme  and 
mean  ratio  by  the  last  problem,  and  suppose 
C  the  point  of  division,  and  A  C  the  greater 
segment. 

From  the  point  B,  with  AC,  the  greater  segment,  as  radius, 
strike  another  arc,  cutting  the  arc  BJ)  in  D.  Join  BJ),  DC,  and 
DA,     The  triangle  ABD  is  the  triangle  required. 

DEMONSTRATION. 

As  AC=BDt  by  construction  ;  and  as  AB  is  to  ud  (7,  as  ^(7  is 
to  BC,  by  the  division  of  AB;  therefore, 

AB:BD: : BD : BC 

Now,  as  the  terms  of  this  proportion  are  the  sides  of  the  two 
triangles  about  the  common  angje,  B,  it  follows,  from  (th.  20,  b.  2), 
that  the  two  triangles,  ABD  and  BDC,  are  equiangular  ;  but  the 
triangle  ABD  is  isosceles  ;  therefore,  BDC  is  isosceles  also,  and 
BD=DC;  hut  BD= AC:  hence,  DC=AC  (ax.  1),  and  the  tri- 
angle A  CD  is  isosceles,  which  gives  the  angle  CDA=A.  But 
the  exterior  angle,  BCD=CDA-\-A,  (th.  11,  b.  1).  Therefore, 
BCD,  or  its  equal  B=CDA-]'A;  or  the  angle  B=2A.  Hence, 
the  triangle  ABD  has  each  of  its  angles,  at  the  base,  double  of 
the  third  angle.     Q.  K  D. 

Scholium.  As  the  two  angles,  at  the  base  of  the  triangle  ABD,  are 
equal,  and  each  double  of  the  angle  A,  it  follows  that  the  sum  of 
the  three  angles  is  Jive  times  the  angle  A.  But  as  the  three  angles 
of  every  triangle  always  make  two  right  angles,  or  1 80  degrees, 
therefore,  the  angle  A  must  be  one-fifth  of  two  right  angles,  or  36 
degrees ;  and  BD  is  a  chord  of  36  degrees,  when  AB  \s  b.  radius 
to  the  circle ;  and  ten  such  chords  would  extend  exactly  round  the 
circle. 

PROBLEM     24. 

Within  a  given  circle  to  inscribe  a  triangle,  equiangular  to  a  given 
triangle. 


88 


GEOMETRY. 


Let  ABC  be  the  circle,  and  ahc 
the  given  triangle.  From  any  point, 
as  A,  draw  the  tangent  EAD  to 
the  given  circle  (problem  1 2). 

From  the  point  Ay  in  the  line 
ADf  make  the  angle  DAC=  the 
angle  h,  (problem  6),  and  the  angle 
UAB=  the  angle  c,  and  join  BO. 

The  triangle  ABO  is  inscribed  in  the  circle;  it  is  equiangular 
to  the  triangle  ahc,  and  is  the  triangle  required. 

Proof,  (th.  12,  b.  3). 


PROBLEM    25, 

To  describe  an  equilateral  and  equiangvlar  pentagon  in  a  given 
cirde. 

1st.  Describe    an    isosceles    tri- 
angle,  abc,  having  each  of  the  equal  ^ 
angles,  b  and  c,  double  of  the  third 
angle,  a,  by  problem  23. 

2d.  Inscribe  the  triangle  AB  C,  in 
the  given  circle,  equiangular  to  the 
triangle  ahc,  by  problem  24;  then 
each  of  the  angles,  B  and  O,  is  double  of  the  angle  A. 

3d.  Bisect  the  angles  B  and  O  by  the  hues  BD  and  OJS, 
(problem  3),  and  join  AU,  EB,  OD,  DA,  and  the  figure  AEBOD 
is  the  pentagon  required. 


DEMONSTRATION 


By  construction,  the  angles  BAO,  ABD,  DBO,  BOE,  EOA, 
are  all  equal;  therefore,  by  scholium  to  th.  9,  b.  3,  the  arc  BO, 
AD,  D  0,  AE,  and  EB,  are  all  equal ;  and  if  the  arcs  are  equal 
the  chords  AE,  EB,  cfec,  are  equal.     Q.  E.  D, 


PROBLEM    26. 

To  describe  an  equiangvlar  and  equilateral  polygon,  of  six  sides,  in 
a  circle. 


BOOK     IV. 


89 


Draw  any  diameter  of  the  circle,  as  AB, 
«nd  from  one  extremity,  B,  draw  BD  equal 
Ui  BC,  the  radius  of  the  circle.  The  arc, 
BD  will  be  one-sixth  part  of  the  whole  cir- 
cumference, and  the  chord  BD  will  be  a 
side  of  the  regular  polygon  of  six  sides. 

In  the  A  CBD,  as  CB=CD,  and  BD 
^=^CB,   by   construction    the    A   is    equilateral,   and   of   course 
equiangular. 

But  the  sum  of  the  three  angles  of  every  A,  is  equal  to  two 
right  angles,  or  to  180  degrees;  and  when  the  three  angles  are 
equal  to  each  other,  each  one  of  them  must  be  60  degrees  ;  but 
60  degrees  is  a  sixth  parth  of  360  degrees,  the  whole  number  of 
degrees  in  a  circle  ;  therefore,  the  arc  whose  chord  is  equal  to  the 
radius,  is  a  sixth  part  of  the  circumference  ;  and  a  polygon  of  six 
equal  sides  may  be  inscribed  in  a  circle,  with  each  side  equal  to  the 
radius. 

Car.  Hence,  as  BD,  is  the  chord  of  60  degrees,  and  equal  to  BO 
or     CD,  we  say  generally, /Ao/  the  chord  of  60  is  equal  to  raditis. 


PROBLEM    27. 

To  find  the  side  of  a  regular  polygon  of  fifteen  sides,  which  may 
he  inscribed  in  any  given  circle. 

Let  CB  be  the  radius  of  the  given  circle, 
and  divide  it  into  extreme  and  mean  ratio 
(problem  22),  and  make  BD  equal  to  CE, 
the  greater  part;  then  BD  will  be  a  side 
of  a  regular  polygon  of  ten  sides  (scholium 
to  problem  23).  Draw  BA=^  to  CB,  and 
it  will  be  a  side  of  a  polygon  of  six  sides. 
Join  DAy  and  that  line  must  be  the  side  of  a  polygon,  which  cor- 
responds to  the  arc  of  the  circle  expressed  by  \y  less  y^^,  of  the 
whole  circumference  ;  or  \ — yU=_*_=J_;  that  is,  one-fifteenth  of 
the  whole  circumference ;  or,  DA  is  a  side  of  a  regular  polygon 
of  16  sides. 


GEOMETRY. 


BOOK    V. 


ON    THE    PROPORTIONALITIES    AND     MEASUREMENT    OF    POLYGONS 
AND     CIRCLES. 


THEOREM     1. 

The  area  of  any  circle  is  equal  to  the  product  of  its  radius  into 
half  of  its  circumference. 

Let  (7-4  be  the  radius  of  the  circle,  and 
AB  a  very  small  portion  of  its  circumference, 
and  CAB  will  be  a  sector;  and  we  may 
conceive  the  whole  circle  made  up  of  a  great 
number  of  such  sectors  ;  and  each  sector 
may  be  as  small  as  we  please ;  and  when 
very  small,  ABy  BJDy  <fec.,  each  one  taken 
separately,  may  be  considered  a  right  line  ;  and  the  sectors  CAB, 
CBD^  (fee,  will  be  triangles.  The  triangle  CAB,  is  measured  by 
the  base,  CA,  multiplied  into  half  the  altitude,  (th.  30,  b.  I)  AB; 
and  the  triangle  CBD  is  measured  by  CB,  or  its  equal,  CA,  in'o 
half  BD:  then  the  area,  or  measure  of  the  two  triangles,  or  sectors, 
is  CA,  multiplied  by  the  half  of  AB,  plus  the  half  of  BD,  and 
so  on  for  all  the  sectors  that  compose  the  circle  ;  therefore,  the 
area  of  the  circle  is  measured  by  the  product  of  the  radius  into  half 
tlie  circumference.     Q.  E.  D. 


THEOREM    2. 

Circumferences  of  circles  are  to  one  another  as  their  radii,  and  their 
areas  are  to  one  another  as  the  squares  of  their  radii. 

Let  CA  be  the  radius  of  a  circle  (see  last  figure),  and  Ca  the 
radius  of  another  circle.  Conceive  them  to  be  placed  upon  each 
other  so  as  to  have  the  same  center. 


BOOK    V.  91 

Let  AB  be  a  certain  definite  portion  of  the  circumference  of  the 
larger  circle,  so  that  m  times  AB  will  represent  that  circumference. 

But  whatever  part  AB  is  of  the  greater  circumference,  the  same 
part  ah  is  of  the  smaller;  for  the  two  circles  have  the  same  number 
of  degrees,  and  of  course  susceptible  of  division  into  the  same 
number  of  sectors.     But  by  proportional  triangles  we  have, 
CAiCa\'.AB:ab 

Multiply  the  last  couplet  by  m  (th.  4,  b.  2),  and  we  have, 
CA  :  Ca  :  :  mAB  :  mah 

That  is,  as  the  radius  of  one  circle  is  to  the  radius  of  the  other,  so 
is  the  circumference  of  the  one  to  the  circumference  of  the  other. 

Q.  K  D. 

To  prove  the  second  part  of  the  theorem,  represent  the  larger 
circle  by  C,  and  the  smaller  by  c;  and  whatever  part  the  sector 
CAB  is  of  the  circle  (7,  the  sector  Cab  is  the  same  part  of  the 
circle  c. 

That  is,        .        C:      c::  CAB    :  Cab 

But,     .        .  CAB  :Oab::(  CAf  :  (  Caf     (th.  22,  b.  2) 

Therefore,    .        C\c      :  :  {CAf  \  {Caf    (th.    6,  b.  2) 

Q.  E.  D. 
Scholium.  1.  Circles  are  to  one  another  as  ihit  squares  of  tlieir 
diameters  ;  for  if  squares  be  described  about  any  two  circles,  such 
squares  will  be  squares  on  the  diameters  of  the  circles.  But  each 
circle  is  the  same  proportional  part  of  its  circumscribed  square  ; 
and  as  like  parts  of  things  have  the  same  proportion  to  each  other 
as  the  wholes  (th.  4,  b.  2);  therefore,  circles  are  to  one  another  as 
the  squares  of  their  diameters. 

Scholium  2.  As  the  circumference  of  every  circle,  great  or 
small,  is  assumed  to  contain  360  degrees,  if  we  conceive  the  cir- 
cumference to  be  divided  into  360  equal  parts,  and  one  such  part 
represented  by  ABy  on  one  circle,  or  ah  on  the  other,  AB  and  ab 
will  be  very  near  straight  lines,  and  the  length  of  such  a  line  as 
AB  will  be  greater  or  less  according  to  the  radius  of  the  circle  ; 
but  its  absolute  length  cannot  be  determined  until  we  know  the 
absolute  relation  between  the  diameter  of  a  circle  and  its 
circumference. 


92  GEOMETRY. 

To  measure  the  circumference  of  a  circle,  or,  to  discover  exactly 
how  many  times,  and  part  of  a  time,  it  is  greater  than  its  diameter, 
is  a  problem  of  some  difficulty,  and  requires  patience  and  care  ; 
and  it  can  only  be  done  approximately;  for  as  far  as  investigations 
have  extended,  the  circumference  of  a  circle  is  incommensurable 
"with  its  diameter. 

To  acquire  a  very  clear  and  dis- 
tinct idea  of  the  ratio  between  the 
diameter  and  circumference  of  a 
circle,  the  pupil  must  commence 
with  first  approximations,  and  pro- 
ceed with  great  deliberation. 

Conceive  a  circle  described  on  the 
radius  CA,  and  in  it  describe  a  regular  polygon  of  six  sides 
(problem  26),  and  each  side  will  be  equal  to  the  radius  CA;  hence 
the  whole  perimeter  of  this  polygon  must  be  six  times  the  radius, 
or  three  times  the  diameter.  Let  CA  bisect  hd  in  a.  Produce  Cb 
and  CW,  and  through  the  point  A^  draw  DB  parallel  to  db;,  DB 
will  then  be  a  side  of  a  regular  polygon  of  six  sides,  described 
about  the  circle,  and  we  can  compute  the  length  of  this  line,  DB, 
as  follows :  The  two  triangles,  Cbd^  and  CBD^  are  equiangular, 
by  construction ;  therefore, 

Caidb:  :  CA:  DB. 

Now,  let  us  assume  CA,  or  Cd,  or  the  radius  of  the  circle,  equal 
unity;  then  db=\,  and  the  preceding  proportion  becomes 
Ca  :1  ::  1  :DB 
In  the  right  angle  triangle  Cad,  we  have, 

Ca^+ad^=Cd^         (th,  36,  b.  1) 
That  is,         .         .         Ca*-1-^=1,  because  Cd=\,  and  ad=^^ 

By  reduction,        .         .        Ca=^»j3,  which  value  of  Ca,  put  in 

the  proportion,  we  have, 

>_  2 

^^3  :  1  :  :  1  :  DB,  or  DB=^~^ 

But  the  whole  perimeter  of  the  circumscribing  polygon  is  six 

2  12  _ 

times  DB;  that  is,  six  times   —=,   or,  -r==4  /3=:6,9282032. 

73  73 


BOOK    V.  83 

Thus  w§  have  shown,  that  when  the  radius  of  a  circle  is  1,  the 
perimeter  of  an  inscribed  polygon  of  six  sides,  is  .  6.000000 
And  of  a  similar  circumscribed  polygon,  is  .  .  6.9282032 
But,  if  we  call  the  diameter  1,  the  perimeter 

of  the  inscribed  polygon  of  six  equal  sides 

will  be, 3.0000000 

And  of  the  circumscribed,  will  be  •         .         .        .  3.4641016 

As  we  would  avoid  all  metaphysical  verbiage  in  science,  and 
come  to  the  point  at  once,  we  lay  it  dovm  as  an  axiom,  that 
when  the  radius  of  a  circle  is  1,  and  of  course  the  diameter  2,  the 
circumference  is  greater  than  6,  and  less  than  6.9282032 ;  and  if 
the  diameter  is  1,  the  circumference  must  be  greater  than  3,  and 
less  than  3.4641016  ;  and  this  we  may  call  the  first  approximation 
to  the  ratio  between  the  diameter  and  circumference  of  a  circle. 

Scholium  3.  As  the  area  of  a  circle  is  numerically  equal  to  the 
radius  multiplied  by  half  the  circumference  (th.  2,  b.  5),  therefore, 
if  we  represent  the  radius  by  B,  and  half  the  circumference  by  7t, 
and  the  area  of  the  circle  by  a,  then  we  shall  have  this  equation : 

JR7t=a 

If  we  now  make  jB=1,  this  equation  gives  7i=a;  that  is,  when 
the  radius  of  a  circle  is  \,  the  half  circumference  is  numerically  equal 
to  the  area.  We  will,  therefore,  seek  the  area  of  a  circle  whose 
radius  is  unity;  and  that  area,  if  found,  will  be  numerically  the 
half  circumference,  and  by  inspecting  the  last  figure,  we  perceive 
that  it  is  perfectly  axiomatic  (the  whole  is  greater  than  a  part), 
that  the  area  of  the  sector  CbAd,  is  greater  than  the  triangle 
Cbd,  and  less  than  the  triangle  CBD;  and  the  area  of  the  whole 
circle  is  greater  than  one  polygon,  and  less  than  the  other.  Find- 
ing the  AREA  of  a  circle  y  or  finding  a  square  which  shall  be  equal  to  a 
circle  of  given  diameter ^  is  known  as  the  celebrated  problem  of  squaring 
the  circle, 

THEOREM    3. 

Given,  the  area  of  a  regular  inscribed  polygon,  and  the  area  of  a 
similar  circumscribed  polygon,  to  find  the  areas  of  a  regular  inscribed 
and  circumscribed  j>olygon  of  double  the  number  of  sides. 


94  GEOMETRY. 

Let  C  be  the  center  of  tKe  circle ;  AB  a 
side  of  the  given  inscribed  polygon;  EF 
parallel  to  AB,  a  side  of  the  circumscribed 
polygon. 

If  AM  be  joined,  and  AR  and  J?§  be 
drawn  as  tangents,  at  A  and  B^  AM  will  be 
a  side  of  an  inscribed  polygon  of  double  the 
number  of  sides;  and  AR=RM  (scholium  2,  th.  18,  b.  3), 
BQ=QM,  and  AR-{'RM=RQ=  the  side  of  the  circumscribed 
polygon  of  double  the  number  of  sides. 

The  As  ARO  and  RMC,  are  equal,  for  AC=CM.  CR  is 
common  to  both  triangles,  and  AR=RM,  tangents  from  the  same 
point,  R;  therefore,  OR  bisects  the  angle  EOM, 

Now,  as  the  same  construction,  and  the  same  reasoning  would 
take  place  at  every  one  of  the  equal  sectors  of  the  circle,  it  is  suf- 
ficient to  consider  one  of  them,  and  whatever  is  true  of  that  arc, 
would  be  true  of  every  one,  and  true  for  the  whole  circle,  and  its 
polygons. 

To  avoid  confusion,  let  p  represent  the  area  of  the  given 
inscribed  polygon,  and  P  the  area  of  the  similar  circumscribed 
polygon.  Also  let  p'  represent  the  area  of  an  inscribed  polygon 
of  double  the  number  of  sides,  and  P'  the  circumscribed  polygon 
of  double  the  number  of  sides. 

As  the  As  AGD  and  ACM  have  the  common  vertex  A,  they 
are  to  each  other  as  their  bases,  CD  to  CM ;  they  are  also  to  each 
other  as  the  polygons  of  which  they  form  a  part. 

Hence,         .         .     p  :  p'  '.  :  CD  \  CM       (1) 

As  AD  and  JSM  are  parallel,  we  have, 

CAiCEw  CD.  CM    (2) 

But,  because  of  the  common  vertex,  J/",  the  two  As,  CAM  and 
CEMy  are  to  each  other  as  CA  to  CE.  But  the  As  are  like  parts 
of  the  polygons  p'  and  P  ;  we  have, 

Therefore,         .  p'  \P\\CA\CE     (3) 

That  is,    .         .  p'  ',P\\CD\  CM     (4)     (th.   6,  b.  2) 

By  comparing  (1)  and  (4),  we  have, 

p'  \P\\p\p\  or p'=: J PXp 


CM:  CE 
CD  :  CA  or  CM 
CD  :  CM 
P  'P' 


BOOK   V.  95 

That  is,  the  area  ofp'  is  a  mean  proportional  between  P  and  p. 
The  two  As,  BMC  and  EEC,  having  the  same  vertex,  C,  are 
to  each  other  as  their  bases,  MH  to  HE, 

But,  because  CH  bisects  the  angle  ECM,     (th.  25,  b.  2) 

MB:  BE 
But,  .  .  CM:  CE 
That  is,  .  BMC:  EBC 

Or,         .  BMC :  EBC 

By  composition,  (th.  8,  b.  2), 

2(BMC) :  (BMC+EBC)  :  :  2p  :p+p' 
But  2  times  BMC  is  P\  and  (BMC+EBC)  is  P 
Therefore,       .         .     P'  :  P  :  :  2p  :p+p' 

Or,         .         .        ,         .    P-?^ 

Now,  P'  is  known,  because  2pP  is  known ;  and  jo-f-jo'  is  also 
known,  as  p'  has  been  previously  determined.  Hence,  by  means 
of  P  and^,  we  can  determine  P'  and^'.     Q.  E.  D, 

Scholium.  By  inspecting  the  figure  in  the  scholium  to  theorem  2, 
we  perceive,  that  if  we  double  the  number  of  sides  of  the  inscribed 
polygon,  we  shall  more  nearly  fill  up  the  circle  ;  and  if  we  double  the 
number  of  sides  of  the  circumscribed  polygons,  we  shall  more  nearly 
pare  them  down  to  the  surface  of  the  circle. 

Hence,  by  continually  increasing  the  sides  of  the  polygons,  as  indi- 
cated by  the  last  theorem,  we  can  find  two  polygons  which  shall  diffier 
from  each  other  by  the  smallest  conceivable  quantity;  but  the  surface 
of  the  circle  is  always  between  the  two  polygons  ;  and  thus  the  sur 
face  of  the  circle  can  be  determined  to  any  assignable  degree  of 
exactness. 

By  taking  the  figure  in  the  scholium  to  theorem  2,  b.  6,  we  perceive 
that  the  area  of  an  inscribed  polygon  of  six  sides,  to  radius  unity 
must   be  .        .         CaXdaX^ 

Which  is        .         .         fV^>  because   da=^ 

And,        .         .         .         Ca2-|-rfa2==Cd'=l 

Or,  ...         Ca=i^Z 

Hence,  .  .  .  i>/3XiX6=fV3=p,  which  corresponda 
with  pt  in  the  last  theorem. 


96  GEOMETRY. 

The  area  of  the  circumscribing  polygon  is  measured  by 

CAX^AX6=6DA=3D5. 

But        .        .        .        Caidb'.'.CA:  DB.        (th.  17,  b.  2.) 

>_  2 

That  is,         .        .     1 J3  :  1   :  :  1  :  DB,   or   BD=-^ 

>/3 
6  _ 

Therefore, .        .       ZDB=  /"q~^  V^*  which  corresponds  with  the 

last  theorem. 

Having,  now,  the  area  of  an  inscribed  and  circumscribed  polyfron 
of  six  sides,  by  applying  the  last  theorem  we  can  readily  determine  the 
area  of  an  inscribed  and  a  circumscribed  polygon  of  12  sides. 

Thus,        .         .  |,'=V7P=VIV3X2^-3=3 

2pP    2X|VTX2VT         18      12  _ 

P'-^P  3+fV3         3-1-IV3     2+^3  ^^ 

Now  let  p'  and  P'  be  the  given  polygons,  and  find  others  of  double 
the  number  of  sides,  and  thus  continue  until  the  inscribed  and  circum 
scribed  so  nearly  coincide,  as  to  determine  a  very  approximate  area  of 
the  circle. 

In  this  manner  we  formed  the  following  table  : 


Number  of  sides. 

Inscribed  polygons. 

Circumscribed  polygons 

6 
12 

24 
48 

3= 

6 

:  2.69807621 

3.0000000 

;  3.1058286 
3.1326287 

2^3= 
12 

=3.46410161 

Q  ni KonnA 

2+V3" 

3.1596602 
3.1460863 

V2+V-3 

96 

3.1393554 

3.1427106 

192 

3.1410328 

3.1418712 

384 

3.1414519 

3.1416616 

768 

3.1415568 

3.1416092 

1536 

3.1415829 

3.1415963 

3072 

3.1415895 

3.1415929 

6144 

3.1415912 

3.1415927 

Thus  we  have  found,  that  when  the  radius  of  a  circle  is  1,  the  semi- 
circumference  must  be  more  than  3.1415912,  and  less  than  3.1415927 ; 
and  this  is  as  accurate  as  can  be  determined  with  the  small  number  of 


BOOK    V.  97 

<fecimals  here  used.  To  be  more  accurate  we  must  have  more  decimal 
places,  and  go  through  a  very  tedious  mechanical  operation  ;  but  this  is 
not  necessary,  for  the  result  is  well  known,  and  is  3.1416926535897 
plus  other  decimal  places  to  the  100th,  without  termination.  This  was 
discovered  through  the  aid  of  an  infinite  series  in  the  differential  and 
integral  calculus. 

The  number  3.1416  is  the  one  generally  used  in  practice,  as  it  is 
much  more  convenient  than  a  greater  number  of  decimals,  and  it  is 
sufficiently  accurate  for  all  ordinary  purposes. 

In  analytical  expressions  it  has  become  a  general  custom  with 
mathematicians  to  represent  this  number  by  the  Greek  letter  rtt  and, 
therefore,  when  any  diameter  of  a  circle  is  represented  by  D,  the  cir- 
cumference of  the  same  circle  must  be  rtD.  If  the  radius  of  a  circle 
18  represented  by  JR,  the  circumference  must  be  represented  by  27iR. 

As  a  farther  discipline  of  mind,  and  for  more  practical  utility,  as 
applicable  to  trigonometry,  we  give  another  method  of  determining  the 
circumference  of  a  circle,  when  the  diameter  is  given.  It  is  evident 
that  when  we  take  a  small  arc,  the  chord  and  the  arc  are  nearly  of  the 
,8ame  length ;  but  the  arc  is  greater  than  the  chord,  for  the  chord  is  a 
straight  line,  and  the  arc  is  curved.  But  if  we  take  the  half  of  any 
emaH  arc,  and  draw  two  chords  in  place  of  one,  such  chords  taken 
together,  will  be  much  nearer  to,  and  more  nearly  equal  in  length  to 
the  arc  than  the  one  chord  of  the  undivided  arc  would  be. 

Now,  if  we  can  divide  the  circumference  into  several  thousand  equal 
parts,  and  can  find  the  length  of  a  chord  corresponding  to  one  of  these 
parts,  the  sum  of  all  these  equal  chords  will  be  infinitely  njear  the  cir- 
cumference of  the  circle  ;  and  the  length  of  such  a  small  chord  we  can 
find,  'provided  we  can  first  know  the  chord  of  any  definite  arc,  and  from 
that  deduce  the  chord  of  any  definite  portion  of  that  arc  ;  and  this  is 
shown  in  the  following  theorem. 

THEOREM     I. 

Given,  the  chord  of  any  arc,  to  determine  the  chord  of  half  that  arc. 

Let  AB  represent  a  given  chord. 
Bisect  the  arc  AB  in  D,  and  join  AD. 
From  C,  the  center  of  the  circle,  draw 
CG  perpendicular  to  AD;  and  from  D, 
draw  DF  perpendicular  to  AB. 

From  AB   we  are  to  determine  AD. 
The  two  Asj  CAn  and  AFD,  are  equi- 
angular ;  for  the  angle  FAD,  at  the  circumference,  is  measured  by 
7 


98  GEOMETRY. 

half  the  arc  BD;  and  nCA,  at  the  center,  is  measured  by  half  of  an 
equal  arc,  AD,    The  right  angle,  F=  the  right  angle  CnA;  therefore, 

As        .        .         ,      DA  :AF  :  :  CA  :  Cn. 

In  the  triangle  CwA,Iet  Cn=y,  nA=x,  and  CA=1. 
Then  AD=2x;  and  put  AB=C;  then  AF=^C. 
By  this  notation  the  preceding  proportion  becomes 

C 

2a?  :  ^C  :  :  1  :  y.    Hence,  y==TZ 

But  in  the  right  angled  triangle  CnA,  we  have 

By  taking  the  value  of  y',  from  the  proportion,  and  reducing,  we  have 
the  quadratic 

By  adding  4  to  both  members  (see  Alg.  Art.  99),  and  extracting 
square  root,  we  have 

4x2— 2=zbV4— C» 


Therefore,        .        .  .  2x=j2 — ^4 q2 

As  2x  is  the  value  of  AD,  the  expression  (2 — J 4: — C')*  is  the 
value  of  the  chord  of  the  half  of  any  arc,  when  C  represents  the 
value  of  the  chord  of  the  whole  arc.  We  must  take  the  mimis  sign  to 
the  part  represented  by  ^4 — C,  as  the  plus  sign  would  give  increas- 
ing, and  not  decreasing  values. 

If  we  represent  the  chord  of  a  given  arc  by  C,  and  the  chord  of  half 

that  arc  by  C^,  and  the  chord  of  half  that  arc  by  C,,  and  the  chord  of 

half  that  arc  again  by  C,,  &c.,  &c.,we  shall  have  the  following  series 

of  equations  : 

C=  the  first  chord 

(2— V4^=cr')*=c, 

(2-.V4lIc;)^=C, 

(2— V4=Cj)i=^« 
&,c.=&c. 

To  apply  these  equations,  we  observe  that  in  any  circle  the  chord  of 
60°  is  equal  to  the  radius  (cor.  to  prob.  26),  and  if  the  radius  is  assumed 
as  unity,  we  have, 

C  =  chord  of  60°  =1.000000000    sid. 

ins.  pol.  of       6  sides. 

(a— ^"iHc*  )*==  ^i  =  chord  of  30°  ■»  .6176380902  sid. 

ins.  pol.  of      12  sides. 


BOOK    V 


09 


(2— ^4— Cf  )*=C,=  chord  of  16° 
ins.  pel.  of      24  sides. 

(2— ^4Z:cf)^=C3=chordof  7°  30' 
ins.  pol.  of      48  sides. 

(2— 74^Zc|  )*=  C^=  chord  of   3°  46 
ins.  pol.  of      96  sides. 


.2610523842  sid. 


.1308062583  sid. 


.0654381655  sid. 


(2-^4-C:)^= 

ins.  pol.  of 


zCg=  chord  of 
192  sides. 


1°  62'  30'  =  .0327234632  sid. 


(2— V4— C|  )^=C^=z  chord  of         66'  16"  =  .0163622792  sid. 

ins.  pol.  of    384  sides. 

(2—^4^=1^6  )^=C',=  chord  of         28'    7"    30'"=  .0081812080  sid. 
ins.  pol,  of    768  sides. 


i*Ni. 


14'    3"  46  '"=  .0040906112  sid. 


=  .0U20463068  sid. 


(2— ^4— C;  y=C^=  chord  of 
ins.  pol.  of  1536  sides, 

(2— J4^irc|  )L  c^=  chord  of  7'  &c. 

-ins.  pol.  of  3072  sides. 

Hence,  .0020453068X3072=6.2831814896,  is  the  perimeter  of  an 
inscribed  polygon  of  3072  sides  when  the  radius  is  I,  or  diameter  2. 
When  the  diameter  is  1,  the  perimeter  is  3.1415907498,  which  is  a 
a  little,  and  but  a  little,  less  than  the  circumference,  as  determined  by 
more  extended  computations. 

Although  not  necessary  for  practical  application,  the  following 
beautiful  theorem  for  the  analytical  tri-section  of  an  arc  will  not  be 
unacceptable. 


THEOREM    5. 

Given,  the  chord  of  any  arc,  to  determine  the  chord  of  one  third  of  such  arc. 

Let  AJEJ  be  the  given  chord,  and  conceive  its 
arc  divided  into  three  equal  parts,  as  represented 
by  AB,  BD,  and  DE. 

Through  the  center  draw  BCG,  and  join  AB' 
The  two  As,  CAB  and  ABF,  are  equiangular; 
for  the  angle  FAB,  being  at  the  circumference, 
is  measured  by  half  the  arc  BE,  which  is  equal 
to  AB,  and  the  angle  BCAt  at  the  center,  is 


100  GEOMETRY. 

measured  by  the  arc  AB;  therefore,  the  angle  FAB=:BCA;  but  the 
angle  CBA  or  FBAj  is  common  to  both  triangles;  therefore,  the  third 
angle,  CAB,  of  the  one  triangle,  is  equal  to  the  third  angle,  AFB, 
of  the  other  (th.  ll,b.  1,  cor.  2),  and  the  two  triangles  are  equiangular 
and  similar. 

But  the   A  CBA  is  isosceles;    therefore,  the   A  AFB    is  also 
isosceles,  and  AB=AF,  and  we  have  the  following  proportions  : 
CAiAB:  :  AB  :  BF 

Now  let  AE—c,  AB=Xt  CA=1.  Then  AF=x,  and  EF=c—Xi 
and  the  proportion  becomes, 

I'.xi'.x:  BF.     Hence  BF=x'^ 

Also,        .        .         .        .FG=2.—aP 

As  AE  and  GB  are  two  chords  that  intersect  each  other  at  the 

point  F,  we  have, 

GFXFB=AFXFE        (th.  17,  b.  3) 

That  is,    .        .  (2—x^)x^=x(c—x) 

Or,   .        .        .        .     a?5 — 3x=—c 

If  we  suppose  the  arc  AF  to  be  60  degrees,  then  c=I,  and  the 
equation  becomes  x^ — 3x= — 1 ;  a  cubic  equation,  easily  resolved  by 
Horner's  method  (  Robinson's  Algebra,  University  Edition,  Art.  193), 
giving  ar=. 347296-]-,  the  chord  of  20°.  This  again  may  be  taken  for 
the  value  of  c,  and  a  second  solution  will  give  the  chord  of  6°  40',  and 
so  on,  trisecting  as  many  times  as  we  please. 

If  the  pupil  has  carefully  studied  the  foregoing  principles,  he  has 
the  foundation  of  all  geometrical  knowledge;  but  to  acquire  indepen- 
dence and  confidence,  it  is  necessary  to  receive  such  discipline  of 
mind  as  the  following  exercises  furnish. 

Some  of  the  examples  are  mere  problems,  some  are  theorems,  and 
some  a  combination  of  both.  Care  has  been  taken  in  their  selection, 
that  they  should  be  appropriate  ;  not  very  severe,  not  such  as  to  try 
the  powers  of  a  professed  geometrician,  nor  such  as  would  be  too 
trifling  to  engage  serious  attention. 

EXERCISES    IN   GEOMETRICAL    INVESTIGATION. 

1.  From  two  given  points,  to  draw  two  equal  straight  lines,  which 
shall  meet  in  the  same  point,  in  a  line  given  in  position. 

2.  From  two  given  points  on  the  same  side  of  a  line,  given  iii 
position  to  draw  two  lines  which  shall  meet  in  that  line,  and  make 
equal  angles  with  it. 

3.  If  from  a  point  without  a  circle,  two  straight  lines  be  drawn  to 


BOOK    V.  101 

the  concave  part  of  the  circumference,  making  equal  angles  with  the 
line  joining  the  same  point  and  the  center,  the  parts  of  these  lines 
which  are  intercepted  within  the  circle,  are  equal. 

4.  If  a  circle  be  described  on  the  radius  of  another  circle,  any 
straight  line  drawn  from  the  point  where  they  meet,  to  the  outer  cir- 
cumference, is  bisected  by  the  interior  one. 

6.  From  two  given  points  on  the  same  side  of  a  line  given  in  posi- 
tion, to  draw  two  straight  lines  which  shall  contain  a  given  angle,  and 
be  terminated  in  that  line. 

6.  If,  from  any  point  without  a  circle,  lines  be  drawn  touching  it 
the  angle  contained  by  the  tangents  is  double  the  angle  contained  by 
the  line  joining  the  points  of  contact,  and  the  diameter  drawn  through 
one  of  them. 

7.  If,  from  any  two  points  in  the  circumference  of  a  circle,  there  be 
drawn  two  straight  lines  to  a  point,  in  a  tangent,  to  that  circle,  they 
will  make  the  greatest  angle  when  drawn  to  the  point  of  contact. 

8.  From  a  given  point  within  a  given  circle,  to  draw  a  straight  line 
which  shall  make,  with  the  circumference,  an  angle,  less  than  any 
angle  made  by  any  other  line  drawn  from  that  point. 

9.  If  two  circles  cut  each  other,  the  greatest  line  that  can  be  drawn 
through  the  point  of  intersection,  is  that  which  is  parallel  to  the  line 
joining  their  centers. 

10.  If.  from  any  point  within  an  equilateral  triangle,  perpendiculars 
be  drawn  to  the  sides,  they  are,  together,  equal  to  a  perpendicular 
drawn  from  any  of  the  angles  to  the  opposite  side. 

11.  If  the  points  of  bisection  of  the  sides  of  a  given  triangle  be 
joined,  the  triangle,  so  formed,  will  be  one-fourth  of  the  given  triangle. 

12.  The  difference  of  the  angles  at  the  base  of  any  triangle,  is  double 
the  angle  contained  by  a  line  drawn  from  the  vertex  perpendicular  to 
the  base,  and  another  bisecting  the  angle  at  the  vertex. 

13.  If,  from  the  three  angles  of  a  triangle,  lines  be  drawn  to  the 
points  of  bisection  of  the  opposite  sides,  these  lines  intersect  each 
other  in  the  same  point. 

14.  The  three  straight  lines  which  bisect  the  three  angles  of  a  tri- 
angle, meet  in  the  same  point. 

16.  The  two  triangles,  formed  by  drawing  straight  lines  from  any 
point  within  a  parallelogram  to  the  extremities  of  two  opposite  sides, 
are,  together,  half  the  parallelogram. 

16.  The  figure  formed  by  joining  the  points  of  bisection  of  the  sides 
of  a  trapezium,  is  a  parallelogram. 

17.  If  squares  be  described  on  three  sides  of  a  right  angled  triangle. 


102  GEOMETRY. 

and  the  extremities  of  the  adjacent  sides  be  joined,  the  triangles  so 
formed,  are  equal  to  the  given  triangle,  and  to  each  other. 

18.  If  squares  be  described  on  the  hypotenuse  and  sides  of  a  right 
angled  triangle,  and  the  extremities  of  the  sides  of  the  former,  and  the 
adjacent  sides  of  the  others,  be  joined,  the  sum  of  the  squares  of  the 
lines  joining  them,  will  be  equal  to  five  times  the  square  of  the 
hypotenuse. 

19.  The  vertical  angle  of  an  oblique-angled  triangle,  inscribed  in  a 
circle,  is  greater  or  less  than  a  right  angle,  by  the  angle  contained 
between  the  base,  and  the  diameter  drawn  from  the  extremity  of 
the  base. 

20.  If  the  base  of  any  triangle  be  bisected  by  the  diameter  of  its 
circumscribing  circle,  and,  from  the  extremity  of  that  diameter,  a  per- 
pendicular be  let  fall  upon  the  longer  side,  it  will  divide  that  side  into 
segments,  one  of  which  will  be  equal  to  half  the  sum,  and  the  other 
to  half  the  difference  of  the  sides. 

21.  A  straight  line  drawn  from  the  vertex  of  an  equilateral  triangle, 
inscribed  in  a  circle,  to  any  point  in  the  opposite  circumference,  is 
equal  to  the  two  lines  together,  which  are  drawn  from  the  extremities 
of  the  base  to  the  same  point. 

22.  The  straight  line  bisecting  any  angle  of  a  triangle  inscribed  in 
a  given  circle,  cuts  the  circumference  in  a  point,  which  is  equidistant 
from  the  extremities  of  the  sides  opposite  to  the  bisected  angle,  and 
from  the  center  of  a  circle  inscribed  in  the  triangle. 

23.  If,  from  the  center  of  a  circle,  a  line  be  drawn  to  any  point  in 
the  chord  of  an  arc,  the  square  of  that  line,  together  with  the  rectangle 
contained  by  the  segments  of  the  chord,  will  be  equal  to  the  square 
described  on  the  radius. 

24.  If  two  points  be  taken  in  the  diameter  of  a  circle,  equidistant 
from  the  center,  the  sum  of  the  squares  of  the  two  lines  drawn  from 
these  points  to  any  point  in  the  circumference,  will  be  always  the  same. 

25.  If,  on  the  diameter  of  a  semicircle,  two  equal  circles  be  described, 
and  in  the  space  included  by  the  three  circumferences,  a  circle  be  in- 
scribed, its  diameter  will  be  |  the  diameter  of  either  of  the  equal 
circles. 

26.  If  a  perpendicular  be  drawn  from  the  vertical  angle  of  any 
triangle  to  the  base,  the  difference  of  the  squares  of  the  sides  is  equal 
to  the  difference  of  the  squares  of  the  segments  of  the  base. 

27.  The  square  described  on  the  side  of  an  equilateral  triangle,  is 
equal  to  three  times  the  square  of  the  radius  of  the  circumscribing 
circle. 


BOOK    V.  iO§ 

28.  The  sum  of  the  sides  of  an  isosceles  triangle,  is  less  than  the 
sum  of  any  other  triangle  on  the  same  base  and  between  the  same 
parallels. 

29.  In  any  triangle,  given  one  angle,  a  side  adjacent  to  the  given 
angle,  and  the  difference  of  the  other  two  sides,  to  construct  the 
triangle. 

30.  In  any  triangle,  given  the  base,  the  sum  of  the  other  two  sicfcs, 
and  the  angle  opposite  the  base,  to  construct  the  triangle. 

31.  In  any  triangle,  given  the  base,  the  angle  opposite  to  the  base, 
and  the  difference  of  the  other  two  sides,  to  construct  the  triangle. 

PROBLEMS    REQUIRING    THE    AID    OF    ALGEBRA 
FOR    THEIR   SOLUTION. 

No  definite  rules  can  be  given  for  the  solution  or  construction  of 
the  following  problems;  and  the  pupil  can  have  no  other  resources 
than  his  own  natural  tact,  and  the  application  of  his  analytical  and 
geometrical  knowledge  thus  far  obtained  ;  and  if  that  knowledge  is 
sound  and  practical,  the  pupil  will  have  but  little  difficulty;  but  if  his 
geometrical  acquirements  are  superficial  and  fragmentary,  the  difficul- 
ties may  be  insurmountable  :  hence,  the  ease  or  the  difficulty  which 
we -experience  in  resolving  such  problems,  is  the  test  of  an  efficient  or 
inefficient  knowledge  of  theoretical  geometry. 

When  a  problem  is  proposed  requiring  the  aid  of  Algebra,  draw  the 
figure  representing  the  several  parts,  both  known  and  unknown.  Rep- 
resent the  known  parts  by  the  first  letters  of  the  alphabet,  and  the 
unknown  and  required  parts  by  the  final  letters,  &c.;  and  use  whatever 
truths  or  conditions  are  available  to  obtain  a  sufficient  number  of 
equations,  and  the  solution  of  such  equations  will  give  the  unknown 
and  required  parts  the  same  as  in  common  Algebra. 

But  as  we  are  unable  to  teach  by  more  general  precept,  we  give  the 
solutions  of  a  few  examples,  as  a  guide  to  the  student. 

The  first  two  are  specimens  of  the  most  simple  and  easy;  the  last 
two  or  three  are  specimens  of  the  most  difficult  and  complex,  or  such 
as  might  not  be  readily  resolved,  in  case  solutions  were  not  given. 

It  might  be  proper  to  observe  that  different  persons  might  draw 
different  figures  to  the  more  complex  problems,  and  make  different 
equations  and  give  different  solutions ;  but  the  best  solutions  are 
always  the  most  simple. 

PROBLEM    1. 

Given,  the  hypotenuse,  and  the  sum  of  the  other  two  sides  of  a  right 
angled  triangle,  to  determine  the  triangle. 


104 


GEOMETRY. 


Let  ABC  be  the  A-  Put  CB=y,A£:=Xy  AC=hy 
and  CB'\-AB=^.  Then,  by  a  given  condition  we 
we  have, 

x~\-y=s 
And,        .         .       x'-f-y2=A'        (th.  36,  b.  1) 
From  these  two  equations  a  solution  is  easily  ob- 
tained, giving, 

If    h=5,  and  5=7,  x=4  or  3,  and  y=3  or  4. 

N.  B.     In  place  of  putting  x  to  represent  one  side,  and  y  the  other, 

we  might  put  (x-\-i/)  to  represent  the  greater  side,  and  (x — y)  the  lesser 

Bide;  then,       «        .   x^-\-y^=-^*    and  2x=5,  &c. 


PROBLEM    2. 

Given,  the  hose  and  perpendicular  of  a  triangle,  to  find  the  side  of  its 
inscribed  square. 

Let  ABC  be  the  A-  AB=ly  the 
base,  CD=Pi  the  perpendicular. 

Draw  EF  parallel  to  AB,  and  suppose 
It  equal  to  jEG,  a  side  of  the  required 
square;  and  put  EF=x. 

Then,  by  proportional  As  we  have, 


CI :  EF  :  :  CD  :  AB 

That  is,    p — X  :    x    :  :    p    :    b 
Hence,         .      bp — bx=px;    or,  x= 


'■b+p 


That  is,  the  side  of  the  inscribed  square  is  eqiud  to  the  product  of  the  base 
and  altitude,  divided  by  their  sum. 

PROBLEM     3. 

In  a  triangle,  having  given  the  sides  about  the  vertical  angle,  and  tht 
ine  bisecting  that  angle  and  terminating  in  the  base,  to  find  the  base. 

Let  ABC  he  the  Aj  and  let  a  circle  be  cir- 
cumscribed about  it.  Divide  the  arc  AEB  into 
two  equal  parts  at  the  point  E,  and  join  EC. 
This  line  bisects  the  vertical  angle  (th.  9,  b.  3, 
scholium).     Join  BE. 

Put  AD=x,  DB=y,  A  C=:a,  CB=zb,  CD=c, 
and  DE=w.  The  two  As,  ADC  and  EBC, 
are  equiangular;  from  which  we  have, 

w-\-c  :  b  :  :  a  :  c;  or,  cw-[-c*=ab        (1) 


v^^^. 


BOOK    V.  lOJi 

But,  BB  EC  and  AB  bi&  two  chords  that  intersect  each  other  in  r* 
circle,  we  have,        ....  cwz=zocy  (th.  17,  b.  3) 

Therefore,  ....       xi/-\-c^=ab  (2) 

But,  as  CD  bisects  the  vertical  angle,  we  have, 
a  :h  :  :x  ly        (th.  23,  b.  2) 

Or,        .  .     a:=f  (3) 


JPf 


a  I        c*h 

Hence,.        ^y*-j-c2=a5j   ory=^6* — — 

And,        ....        x=-^b^-^-^ 

Now,  as  X  and  y  are  determined,  the  base  is  determined. 

N.  B.     Observe  that  equation  (2)  is  theorem  20,  book  3. 


PROBLEM    I. 

^o  determirie  a  triangle,  from  the  base,  the  line  bisecting  the  vertical 
cmgle,  tend  the  diameter  of  the  circumscribing  circle. 

Describe  the  circle  on  the  given  diameter, 
AB,  and  divide  it  in  two  parts,  in  the  point  D, 
BO  that  ADxDB  shall  be  equal  to  the  square 
of  one  half  the  given  base. 

Through  D  draw  EDG  at  right  angles  to 
ABj  and  EG  will  be  the  given  base  of  the 
triangle. 

Put     .  AD=n,  DB=my  AB=:d,  DG=b. 

Then,  n-\-m=d,  and  nm=b* ;  and  these  two  equations  will  deter* 
mine  n  and  m;  and  therefore,  n  and  m  we  shall  consider  as  known. 

Now,  suppose  EHG  to  be  the  required  A>  and  join  HIB  and  HA . 
The  two  As,  AHB,  DBI,  are  equiangular,  and  therefore,  we  have, 
AB  :  HB  :  :  IB  :  DB. 

But  HI  is  a  given  line,  that  we  will  represent  by  c;  and  if  we  put 
IB=Wi  we  shall  have  HB=c-\-w;  then  the  above  proportion  becomes, 
d  :  c-\-w  :  '.to  :m 

Now,  w  can  be  determined  by  a  quadratic  equation;  and  therefore, 
IB  is  a  known  line. 

In  the  right  angled  A  DBI,  the  hypotenuse  IB,  and  base  DB,  are 
known;  therefore,  DI  is  known  (th.  36,  b.  1);  and  if  DI  is  known, 
EI  and  IG  are  known. 


106  GEOMETRY. 

Lastly,  let  EH=x,  HG=y,  and  put  EI=p,  and  IG=q. 

Then,  by  theorem  20,  book  3,        pq-\-c*=xy        (1) 

But, X  :y  :  :p  :q  (th.  25,  b.  2) 

Or, x=^        (2) 

And,  from  equations  (1)  and  (2)  we  can  determine  x  and  y,  the  sides 
of  the  A;  and  thus  the  determination  has  been  attained,  carefully  and 
easily,  step  by  step. 

PROBLEM    5. 

Three  equal  circles  touch  each  other  externally ^  and  thus  inclose  one  acre 
of  ground;  what  is  the  diameter  in  rods  of  each  of  these  circles  ? 

Draw  three  equal  circles  to  touch  each  other 
externally,  and  join  the  three  centers,  thus 
forming  a  triangle.  The  lines  joining  the 
centers  will  pass  through  the  points  of  con- 
tact (th.  7,  b.  3). 

Let  R  represent  the  radius  of  these  equal 
circles ;  then  it  is  obvious  that  each  side  of  this 
A  is  equal  to  2R.  The  triangle  is  therefore 
equilateral,  and  it  incloses  the  given  area,  and  three  equal  sectors. 

As  each  sector  is  a  third  of  two  right  angles,  the  three  sectors  are, 
together,  equal  to  a  semicircle;  but  the  area  of  a  semicircle,  whose 

ftR^ 
radius  is  U,  is  expressed  by  — r-    (th.  3,  b.  5,  and  th.  1,  b.  5);  and  the 

ftR^ 
area  of  the  whole  triangle  must  be    —^-{-160;  but  the  area  of  the 

A  is  also  equal  to  R  multiplied  by  the  perpendicular  altitude,  which 

IbRJz. 

rcR^ 

Therefore,      .      R*JZ=-2~'^^^^ 


Or,        .    ll«(2V3— '<)=320 
320 

2^3— 3.14161 
Hence,         12=31.484-  rods  for  the  result. 


320                 3.20 
Ut_.___ ^ =992.248 

2^3—3.1416926     0.3225 


PROBLEM    6. 

In  a  right  angled  triangle^  having  given  the  base  and  the  sum  of  ihs 
perpendicular  and  hypotenuse,  to  find  these  two  sides. 


BOOK    V.  107 

PROBLEM    7. 

Giveitt  the  hose  and  cdtitude  of  a  trianple,  to  divide  it  into  three  equal 
parts,  by  lines  parallel  to  the  base. 

PROBLEM     8. 

Jn  any  equilateral  A>  given  ike  length  of  the  three  perpendiculars  drawn 
from  any  point  within,  to  the  three  sides,  to  determine  the  sides. 

PROBLEM     9. 

In  a  right  angled  triangle,  having  given  the  base  (3),  ajid  the  difference 
betioeen  the  hypotenuse  and  perpendiciUar  (1),  to  find  both  these  two  sides. 

PROBLEM     10. 

In  a  right  angled  triangle,  having  given  the  hypotenuse  (5),  and  the  dif' 
ference  between  the  base  and  perpendicular  (1),  to  determine  both  these  ttoo 
sides, 

PROBLEM     11. 

Having  given,  the  area  or  measure  of  the  space  of  a  rectangle  inscribed 
in  a  given  triangle,  to  determine  the  sides  of  the  rectangle. 

PROBLEM     12. 

In  a  triangle,  having  given  the  ratio  of  the  two  sides,  together  with  both 
the  segments  of  the  base,  made  by  a  perpendicular  from  the  vertical  angle,  to 
determine  the  sides  of  the  triangle. 

PROBLEM     13. 

In  a  triangle,  having  given  the  base,  the  sum  of  the  other  two  sides,  and 
the  length  of  a  liv£  drawn  from  the  vertical  angle  to  the  middle  of  the  base, 
to  find  the  sides  of  the  triangle. 

PROBLEM     14. 

To  determine  a  right  angled  triangle;  having  given  the  lengths  of  two 
lines  dravmfrom  the  acute  angles  to  the  middle  of  the  opposite  sides. 

PROBLEM     15. 

To  determine  a  right  angled  triangle;  having  given  the  perimeter,  and  the 
radius  of  its  inscribed  circle. 

PROBLEM     16. 

?  To  determine  a  triangle;  having  given  the  base,  the  perpendicular,  and 
the  ratio  of  the  two  sides. 

PROBLEM     17. 

To  determine  a  right  angled  triangle;  having  given  the  hypotenuse,  and 
the  side  of  the  inscribed  square. 


108  GEOMETRY. 

PROBLEM     18. 

To  determine  the  radii  of  three  equal  circles  j  inscribed  in  a  given  circle, 
to  touch  each  other ^  and  also  the  circumference  of  the  given  circle, 

PROBLEM     19. 

In  a  right  angled  triangle,  having  given  the  perimeter,  or  sum  of  aU  the 
sides,  and  the  perpendicular  Ut  fall  from  the  right  angle  on  the  hypotenuse, 
to  determine  the  triangle;  that  is,  its  sides, 

PROBLEM     20. 

To  determine  a  right  angled  triangle;  having  given  the  hypotenuse  and 
the  difference  of  two  lines,  drawn  from  the  two  acute  angles  to  the  center  of 
the  inscribed  circle, 

PROBLEM     21. 

To  determine  a  triangle;  having  given  tJte  base,  the  perpendicular,  and 
the  difference  of  the  two  other  sides. 

PROBLEM     22. 

To  determine  a  triangle;  having  given  the  base,  the  perpendicular,  and 
the  rectangle,  or  product  of  the  two  sides, 

PROBLEM     23. 

To  determine  a  triangle;  having  given  the  lengths  of  three  lines  drawn 
from  the  three  angles  to  the  middle  of  the  opposite  sides. 

PROBLEM     24. 

In  a  triangle,  having  given  all  the  three  sides,  to  find  the  radius  of  the 
inscribed  circle. 

PROBLEM     25. 

To  determine  a  right  angled  triangle;  having  given  the  side  of  the  in- 
scribed square,  and  the  radius  of  the  inscribed  circle, 

PROBLEM     26. 

To  determine  a  triangle,  and  the  radius  of  the  inscribed  circle;  having 
given  the  lengths  of  three  lines  drawn  from  the  three  angles  to  the  center  of 
that  circle. 

PROBLEM     27. 

To  determine  a  right  angled  triangle;  having  given  the  hypotenuse,  and 
thi  radius  of  the  inscribed  circle. 


BOOK    VI.  109 


BOOK     YI. 


ON     THa     IHTBRSBCTION     OF     PLJLNXS. 

DEFINITIONS. 

The  14tli  definition  of  book  t,  defines  a  plane.  It  is  a  superfices, 
having  length  and  breadth,  but  no  thickness. 

The  surface  of  still  water,  the  side  of  a  sheet  of  paper,  may 
give  a  person  some  idea  of  a  plane. 

A  curved  surface  is  not  a  plane  ;  although  we  sometimes  say, 
**  the  plane  of  the  earth's  surface." 

1.  If  any  two  points  he  taken  in  a  plane y  and  a  straight  line  join 
the  points,  every  point  in  that  line  is  in  the  plane, 

2.  If  any  point  in  such  a  line  should  be  either  above  or  below 
the  surface,  such  a  surface  would  not  be  a  plane. 

3.  A  straight  line  is  perpendicular  to  a  plane,  when  it  makes 
right  angles  with  every  straight  line  which  it  meets  in  that  plane. 

4.  Two  planes  are  perpendicular  to  each  other  when  any  straight 
line  drawn  in  one  of  the  planes,  perpendicular  to  their  common 
section,  is  perpendicular  to  the  other  plane. 

5.  If  two  planes  cut  each  other,  and  from  any  point  in  the  line  of 
their  common  section,  two  straight  lines  be  drawn,  at  right  angles 
to  that  line,  one  in  the  one  plane,  and  the  other  in  the  other  plane, 
the  angle  contained  by  these  two  lines  is  the  angle  made  by  the 
planes. 

6.  A  straight  line  is  parallel  to  a  plane  when  it  does  not  meet  the 
plane,  though  produced  ever  so  far. 

7.  Planes  are  parallel  to  each  other  when  they  do  not  meet, 
though  produced  to  any  extent. 

8.  A  solid  angle  is  one  which  is  formed  by  the  meeting,  in  one 
point,  of  more  than  two  plane  angles,  which  are  not  in  the  same 
plane  with  each  other. 


110 


GEOMETRY. 


THEOREM     1. 

If  any  three  straight  lines  meet  erne  another ^  they  are  in  one  plane* 

For  conceive  a  plane  passing  through  BC 
to  revolve  about  that  line  till  it  pass  through 
the  point  E.  Then  because  the  points  E 
and  C  are  in  that  plane,  the  line  EC  is  in 
it ;  and  for  the  same  reason,  the  line  £JB 
is  in  it;  and  BC  is  in  it,  by  hypothesis. 
Hence  the  Unes  AB,  CD^  and  BC  are  all  in 
one  plane. 

Cor.  Any  two  straight  lines  which  meet  each  other,  are  in  one 
plane  ;  and  any  three  points  whatever,  are  in  one  plane. 


THEOREM    2. 

J^  two  planes  cut  one  another y  the  line  of  their  common  section  is 
a  straight  line. 

For  let  B  and  D,  any  two  points  in  the  line 
of  their  common  section,  be  joined  by  the 
straight  line  BD ;  then  because  the  points 
B  and  D  are  both  in  the  plane  AU,  the  whole 
line  BD  is  in  that  plane ;  and  for  the  same 
reason  BJ)  is  in  the  plane  CF.  The  straight  line  BD  is  therefore 
common  to  both  planes  ;  and  it  is  therefore  the  line  of  their 
common  section. 


PROPOSITION    3.      THEOREM. 

^  a  straight  line  stand  at  right  angles  to  each  of  two  other  straight 
lines  at  their  point  of  intersection,  it  mil  be  at  right  angles  to  the  plane 
of  those  lines. 

Let  AB  stand  at  right  angles  to  BF  and 
CD,  at  their  point  of  intersection  A.  T/ien 
AB  will  be  at  rigid  angles  to  any  other  line  drawn 
through  A  in  the  plane,  passing  through  FF, 
CD,  and,  of  course,  at  right  angles  to  the  plane 
itself.     (Def.  3.) 

Through  A,  draw  any  line,  A  Q,  in  the  plane 


BOOK    VI.  Ill 

JEF  CDy  and  from  any  point  G,  draw  GH  parallel  to  AD.  Take 
HF=zAHy  and  join  FG  and  produce  it  to  D.  Because  HG  is 
parallel  to  AD,  we  have 

FH'.HAwFG :  GD 

But,  in  this  proportion,  the  first  couplet  is  a  ratio  of  equality ; 
therefore  the  last  couplet  is  also  a  ratio  of  equality, 

That  is,  FG=  GD,  or  the  line  FD  is  bisected  in  G, 

3om£D,BG,a.nd£F. 

Now,  in  the  triangle  AFD,  as  the  base  FD  is  bisected  in  G, 
we  have,       .       AF^-\-AD'=2AG'-{-2GF'     (1)    (th.  39  b.  1.) 

Also,  as  DF  is  the  base  of  the  A  BDFy  we  have  by  the  same 
theorem,       .       BF^^-BD^^'iBG^^^GF^     (2) 

By  subtracting  (1)  from  (2)  and  observing  that  BF^-—AF^ 
:=AB^  because  ^^i^  is  a  right  angle  ;   and  BD^^AD^=AB\ 
because  BAD  is  a  right  angle,  and  we  shall  then  have, 
AB'-\-AB'=2BG^—^AG'' 

Dividing  by  2,  and  transposing  AG^,  and  we  have, 
AB^-\-AG^=BG^ 

This  last  equation  shows  that  BA  6^  is  a  right  angle.  But  A  G 
.s  any  line  drawn  through  Ay  in  the  plane  FF,  CD,  therefore  AB 
is  at  right  angles  to  any  line  in  the  plane,  and,  of  course,  at  right 
angles  to  the  plane  itself.     Q.  E.  D, 

PROPOSITION  I.  PROBLEM  AND  THEOREM. 

To  draw  a  straight  line  perpendicvlar  to  a  plane,  from  a  given  point 
above  it. 

Let  J/2Vbe  the  plane,  and  A  the  point 
above  it.  Take,  DC,  any  line  on  the 
plane,  and  draw  AC ^i  right  angles  to  it. 

From  the  point  C,  draw  CB  on  the 
plane,  at  right  angles  to  the  line  DC. 

Lastly,  from  Ay  draw  AB  at  right  an- 
gles to  the  line  BCy  and  join  BD.  ABC 
is  a  right  angle  hy  construction,  and  now  if  we  can  prove  that  ABD 
is  also  a  right  angle,  then  AB  is  ai  right  angles  to  the  plane,  hy  the 
last  proposition. 


112  GEOMETRY. 

Because  ABC  is  a  right  angle,  we  have, 

To  both  members  of  this  equation,  add  DC^  and  we  have, 
AB'-^{BC^'\-DC^)=AC'-]-J)C^ 

Because  BCD  is  a  right  angle,  BC^-\-DC^=BD^,  and  because 
ACD'\s2i  right  angle,  AC'^-\-DC^=AD'^,  and  taking  these  latter 
values  in  the  last  equation,  we  have, 

AB'-^-BD'^^AI)^ ;  which  shows  that  ABD 
is  a  right  angle,  and  proves  our  proposition.     Q.  E.  D, 

PROPOSITION    5.     THEOREM. 

Two  straight  lines,  having  the  same  inclination  to  a  plane,  whether 
perpendicular  or  oblique,  are  parallel  to  one  another. 

This  proposition  is  axiomatic  from  our  definition  of  parallel  lines  ; 
for  a  stationary  plane  can  have  but  one  position,  and  the  same  in- 
clination from  any  fixed  position,  must,  of  course,  give  parallel 
lines ;  but,  for  the  sake  of  perspicuity,  we  will  give  the  following 
as  a  demonstration. 

Let  MN  be  a  plane,  and  AB  and  CD  lines 
having  the  same  inclination  to  it. 

Then  AB  and  CD  are  parallel. 

If  the  lines  do  not  meet  the  plane,  produce 
them  until  they  do  meet  it  in  B  and  D. 
Join  the  points  B  and  D,  by  the  line  BD,  and  produce  it  to  II, 

The  angle  CDE=iABD,  otherwise  the  two  lines  would  not  have 
the  same  inclination  to  the  plane.  But  when  one  line,  as  BE,  cuts 
two  others,  as  AB  CD,  making  the  exterior  angle,  CDE,  equal  to 
the  interior  and  opposite  angle  on  the  same  side,  ABE,  then  the 
two  lines,  AB  and  (7i>,  are  parallel.     (Converse  of  th.  6,  b.  1). 

Q.  E,  D, 

k 
PROPOSITION    6.     THEOREM. 

If  itffo  straight  lines  he  dravm  in  any  position  through  parallel 
planes,  they  vjill  he  cut  proportumaUy  hy  the  planes. 


BOOK     VI 


113 


Conceive  three  planes  to  be  parallel,  as 
represented  in  the  figure,  and  take  any  points, 
A  and  JB,  in  the  first  and  third  planes,  and 
join  ABf  which  passes  through  the  second 
plane  at  U, 

Also,  take  any  other  two  points,  as  C  and 
2),  in  the  first  and  third  planes,  and  join 
CDf  the  line  passing  through  the  second 
plane  at  F. 

Join  the  two  lines  by  the  diagonal  AD,  which  passes  through 
the  second  plane  at  Q.  Join  BD,  E0-,  OF,  and  AO,  We  are 
now  to  show  that,  AE  \  EB  \\  GF  \  FD 

For  the  sake  of  perspicuity,  put  A  0=X,  and  GD=i  Y, 

As  the  planes  are  parallel,  BD  is  parallel  EO;  then,  in  the  two 

triangles  ABD  and  AEG,  we  have,  (th.  17  b.  2). 
AE:EB'.:X:Y 
Also,  as  the  planes  are  parallel,  GFh  parallel  to  AC,  and  we 

have,        .        .  CF:FD','.X:Y 

By  comparing  the  proportions,  and  applying  theorem  6,  book  2, 
we  have,       .        .      AE  \  EB  :  :  CF :  FD.     Q.  E.  D. 


PROPOSITION    7.     THEOREM 


If  a  straight  line  be  perpendicular  to  a  plane,  all  planes  passing 
ikrotcgk  thai  line  will  be  perpendicular  to  the  first-mentioned  plane. 

Let  MN'hG  a  plane,  and  AB  perpen- 
dicular to  it.  Let  BC  he  any  other 
plane,  passing  through  AB  ;  this  plane 
will  be  perpendicular  to  MX, 

Let  BD  be  the  common  intersection 
of  the  two  planes,  and  from  the  point  B, 
draw  BE  at  right  angles  to  DB. 

Then,  as  -^4^  is  perpendicular  to  the  plane  MX,  it  is  perpendic- 
ular to  every  line  in  that  plane,  passing  through  B  (def.  3,  b.  6); 
therefore,  ABE  is  a  right  angle.  But  the  angle  ABE  (def.  5, 
b.  6),  measures  the  inclination  of  the  two  planes  ;  therefore,  the 
plane  CB  is  perpendicular  to  the  ulane  MX,  and  thus  we  can  show 


8 


/  ^  or  THE  X 

f   UNIVERSITY  I 


r/si  icnR 


°'  H^ 


A  -A-jA^' 


lt4  GEOMETRY. 

that  any  other  plane,  passing  through  ABy  will  be  perpendicular 
to  MN;  therefore,  <fec.     Q.  E.  D, 

PROPOSITION    8.     THEOREM. 

From  the  same  point  in  a  plane,  hut  one  perpendicular  can  he  erected 
from  the  plane. 

Let  MN  be  a  plane,  and  B  a  point  in  it, 
and,  if  possible,  let  two  perpendiculars,  BA 
and  BC,  be  erected. 

Let  BD  be  drawn  on  the  plane  MK^  coin- 
ciding in  direction  with  the  plane  passing 
through  these  two  perpendiculars. 

Now,  as  a  perpendicular  to  a  plane  is  at  right  angles  to  every 
line  that  can  be  drawn  on  the  plane,  through  the  foot  of  the  per- 
pendicular, therefore,  ABD  is  a  right  angle,  also  CBD  is  a  right 
angle. 

Hence,  ABD=  CBD;  the  greater  equal  to  the  less,  which  is 
absurd  ;  therefore,  BC  must  coincide  with  BA,  and  be  one  and 
the  same  line  ;  therefore,  from  the  same  point,  (fee.     Q.  E.  D. 

PROPOSITION    9.     THEOREM. 

If  two  planes  are  per2:)endicular  to  a  third  plane,  the  common  inter- 
section of  the  two  planes  wUl  he  perpendicular  to  the  third  plane. 

Let  CB  and  BD  be  two  planes,  both  per- 
pendicular to  the  third  plane,  MX,  and  let  B 
be  the  common  point  to  all  three  of  the  planes. 
From  B,  draw  BA  at  right  angles  to  FB ; 
BA  will  be  in  the  plane  BD.  From  B,  draw  also  a  perpendicular 
to  GB,  this  will  be  BA ;  or,  there  may  be  two  perpendiculars 
erected  from  the  same  point,  which  is  impossible  ;  therefore,  BA 
is  a  common  section  to  the  two  planes  J5(7and  BD,  and  it  is  at 
right  angles  to  the  two  lines  BF  dindi  BO,  on  the  plane  MX.  AB 
is  therefore  perpendicular  to  that  plane.   (Prop.  3,  b.  6).    Q.  E.  D. 

PROPOSITION     10.     THEOREM. 

If  a  solid  angle  he  formed  hy  three  plane  angles,  the  sum  of  any 
two  of  them  is  greaXer  than  the  third. 


BOOK    VI.  115 

Let  the  three  angles,  BAD,  DAC,  BAC, 
form  the  solid  angle  A,  The  sum  of  any  two 
of  these  is  greater  than  the  third.  When 
these  angles  are  all  equal,  it  is  evident  that  the 
sum  of  any  two  is  greater  than  the  third,  and  the  proposition 
needs  demonstration  only  when  one  of  them,  as  BA  (J,  is  greater 
than  either  of  the  others  ;  we  are  then  to  prove  that  it  is  less  than 
their  sum. 

On  the  line  A  By  take  any  point,  B,  and  draw  any  line,  as  BD. 
From  the  same  point,  jB,  make  the  angle  ABC=ABD,  and  join 
J)  C.  From  the  point  A,  and  on  the  plane  BA  C,  draw  the  angle 
BAE=BAD.  Now  the  two  plane  triangles  BAD  and  BAE, 
have  a  common  side,  AB,  and  the  angles  adjacent  equal  (th.  14, 
b.  1);  therefore,  the  two  As  are,  in  all  respects,  equal;  and 
AD=:AE,  and  BD=BE. 

In  the  triangle  BD  C,       B  C<CBD+D  G 

But,  .  .        .     BE=:BD 

By  subtraction,    .         .    EC<pC 

In  the  two  triangles,  DAG  and  EAG,  DA=AE,  and  AG  is 
common,  but  EG  is  less  than  GD;  therefore,  the  angle  DA  (7,  op- 
posite DC,  is  greater  than  the  angle  EAG,  opposite  EC.  (Con- 
verse of  th.  A,  b.  1). 

That  is,  .        .      DAG y EAG 

But,       .         .  DAB=BAE 

By  addition,       DA  C-\-DAByBA  G.     ( Ax.  2).     Q.  E.  D. 

PROPOSITION     II.    THEOREM. 

The  sum  of  any  plane  angles  forming  any  solid  angle,  is  always 
less  than  four  right  angles. 

Let  the  planes  which  form  the  solid  angle 
at  A,  be  cut  by  another  plane,  which  we  may 
call  the  plane  of  the  base,  BCDE.  Take 
any  point,  a,  in  this  plane,  and  join  aB,  aG, 
aD,  aE,  &c.,  thus  making  as  many  triangles 
on  the  plane  of  the  base,  as  there  are  trian- 
gular planes  forming  the  solid  angle  A,  But 
as  the  sum  of  the  angles  of  every  A  is  two 


116  GEOMETRY. 

right  angles,  the  sum  of  all  the  angles  of  the  As  which  have  their 
vertex  in  A,  is  equal  to  the  sum  of  all  angles  of  the  As  vrhich 
have  their  vertex  in  a.  But  the  angles  JBCA-{-ACD,  are,  to- 
gether, greater  than  the  angles  £Ca-{-aCD,  or  BCD,  by  the  last 
proposition.  That  is,  the  sum  of  all  the  angles  at  the  bases  of 
the  As  which  have  their  vertex  in  A,  is  greater  than  the  sum  of 
all  the  angles  at  the  bases  of  the  As  which  have  their  vertex  in  a. 
Therefore,  the  sum  of  all  the  angles  at  a,  is  greater  than  the  sum 
of  all  the  angles  at  A,  but  the  sum  of  all  the  angles  at  a,  is  equal 
to  four  right  angles  ;  therefore,  the  sum  of  all  the  angles  at  ^,  is 
less  than  four  right  angles.     Q.  E.  D. 


PROPOSITION     12.     THEOREM. 

If  two  solid  angles  are  formed  hy  three  plane  angles  respectively 
equal  to  each  other,  the  plants  which  contain  the  equal  angles  wUl  be 
equally  inclined  to  each  other. 

Let  the  angle  ASC=DTF, 
and  the  angle  ASB  =  DTE ; 
also  the  angle  BSC=ETF; 
then  will  the  inclination  of  the 
planes,  ASO,  A  SB,  be  equal 
to  that  of  the  planes  DTF, 
DTE. 

Having  taken  SB  at  pleasure,  draw  B  0  perpendicular  to  the 
plane  ASC;  from  the  point  0,  at  which  that  perpendicular  meets 
the  plane,  draw  OA,  OC,  perpendicular  to  aS'^,  SC;  join  AB, 
BC;  next  take  TE=SB;  draw  EP  perpendicular  to  the  plane 
DTF;  from  the  point  P,  draw  PD,  PF,  perpendicular  to  TD, 
TF;  lastly,  join  DE,  EF. 

The  triangle  SAB,  is  right  angled  at  A,  and  the  triangle  TDE, 
at  D;  and  since  the  angle  ASB=DTE,  we  have  SBA  =  TED. 
Likewise,  SB^=TE;  therefore,  the  triangle  SAB  is  equal  to  the 
triangle  TDE;  hence,  SA=TD,  and  AB=DE.  In  like  manner 
it  may  be  shown  that,  SC=TF,  and  BC=EF.  That  granted, 
the  quadrilateral  SA  0  C,  is  equal  to  the  quadrilateral  TDPF; 
ioix.,  place  the  angle  ASO,  upon  its  equal  DTF ;  because  SA=TD, 
and  SC=TF,  the  point  A  will  fall  on  D,  and  the  point  C  on  F; 


BOOK    VI.  117 

and,  at  the  same  time,  A  0,  which  is  perpendicular  to  SA^  will 
fall  on  PI>t  which  is  perpendicular  to  TD^  and,  in  like  manner, 
OC  on.  PF;  wherefore,  the  point  0  will  fall  on  the  point  P,  and 
A  0  will  be  equal  to  DP,  But  the  triangles  A  OB,  BPE,  are 
right  angled  at  0  and  P;  the  hypotenuse  AB=DEy  and  the 
side  A  0=^DP;  hence,  those  triangles  are  equal ;  hence,  the  an- 
gle OAB^PDE.  The  angle  OAB  is  the  inclination  of  the  two 
planes  ASB,  ASC;  the  angle  PDE,  is  that  of  the  two  planes 
DTEy  DTF;  consequently,  those  two  inclinations  are  equal  to 
each  other.     Hence,  If  two  solid  angles  are  formed y  <&c. 

Scholium.  The  angles  which  form  the  solid  angles  at  S  and  T, 
may  be  of  such  relative  magnitudes,  that  the  perpendiculars,  B  0 
and  EP,  may  not  fall  within  the  bases,  ASC  and  DTF;  but  they 
will  always  either  fall  on  the  bases  or  on  the  planes  of  the  bases 
produced,  and  0  will  have  the  same  relative  situation  to  Ay  S,  and 
(7,  as  P  has  to  D,  Ty  and  P.  But,  in  case  that  0  and  P  fall 
on  the  planes  of  the  bases  produced,  the  angles  BCO  and  EFP, 
would  be  obtuse  angles  ;  but  the  demonstration  of  the  problem 
would  not  be  varied  in  the  least. 


lis  GEOMETRY. 


BOOK    VII. 

SOLID    GEOMETRY. 

The  object  of  Solid  Geometry  is  to  estimate  and  compare  the 
surfaces  and  magnitudes  of  solid  bodies  ;  and,  like  Plane  Geometry, 
it  must  rest  on  definitions  and  axioms. 

To  the  definitions  already  given,  we  add  the  following,  as  being 
exclusively  applicable  to  Solid  Geometry. 

Surfaces  are  measured  by  square  units;  so  solids  are  measured 
by  cube  units. 

1.  A  Cube  is  a  solid,  bounded  by  six  equal  square  sur-  WS^BL 
faces,  forming  eight  equal  solid  angles.  iRBH 

All  other  solids  are  referred  to  a  imit  of  this  figure  JbSSm 
for  measurement. 

2.  A  Prism  is  a  solid,  whose  ends  are  parallel,  equal,  and  form 
equiangular  plane  figures ;  and  its  sides,  connecting  these  ends, 
are  parallelograms. 

3.  A  prism  takes  particular  names  according  to  the  figure  of  its 
base  or  ends,  whether  triangular,  square,  rectangular,  pentagonal, 
hexagonal,  <fec. 

4.  A  right  or  upright  prism,  is  that  which  has  the  planes  of 
the  sides  perpendicular  to  the  planes  of  the  ends  or  base. 

6.  A  Parallelopipedon  is  a  prism  bounded  by  six 
parallelograms,  every  opposite  two  of  which  are  equal, 
alike,  and  parallel. 

6.  A  rectangular  parallelopipedon,  is  that  whose  boimding 
planes  are  all  rectangles,  which  are  perpendicular  to  each  other. 

A  rectangular  parallelopipedon  becomes  a  cube  when  all  its  planes 
are  equal. 

7.  A  Cylinder  is  a  round  prism,  having  circles  for  its  |BB|S 
ends  ;  and  is  conceived  to  be  formed  by  the  rotation  of  ■^^■l 
a  right  line  about  the  circumferences  of  two  equal  and  |^^H| 
parallel  circles,  always  parallel  to  the  axis.  I^^SI 

8.  The  axis  of  a  cylinder,  is  the  right  line  joining  the  ■■■■ 


BOOK    VII.  119 

centers  of   the  two  parallel  circles,  about  which  the  figure  is 
described. 

9.  A  Pyramid  is  a  solid,  whose  base  is  any  right  lined  HHHj 
plane  figure,  and  its  sides  triangles,  having  all  their  ver-  HHHH 
tices  meeting  together  in  a  point  above  the  base,  called  ■MHB 
the  vertex  of  the  pyramid.  EB^9 

10.  A  pyramid,  like  the  prism,  takes  particular  names 
from  the  figure  of  the  base. 

11 .  A  Cone  is  a  convex  pyramid,  having  a  circular 
base,  and  is  conceived  to  be  generated  by  the  rotation  of 
a  right  line  about  the  circumference  of  a  circle,  one  end 
of  which  is  fixed  at  a  point  above  the  plane  of  that 
circle. 

12.  The  axis  of  a  cone  is  the  right  fine  joining  the  vertex,  or 
fixed  point,  and  the  center  of  the  circle  about  which  the  figure  is 
described. 

13.  Similar  cones  and  cylinders,  are  such  as  have  their  altitudes 
and  the  diameters  of  their  bases  proportional. 

14.  A  Sphere  is  a  solid,  having  but  one  surface,  which  is  in 
every  part  equally  convex ;  and  every  point  on  such  a  surface  is 
equally  distant  from  a  certain  point  within,  called  the  center. 

15.  A  sphere  may  be  conceived  as  having  been  generated  by  the 
revolution  of  a  semicircle  about  its  axis. 

The  diameter  of  such  a  semicircle  is  the  diameter  of  the  sphere; 
and  the  center  of  the  semicircle  is  the  center  of  the  sphere. 

16.  The  altitude  of  any  solid  is  the  perpendicular  distance  be- 
tween the  parallel  planes,  one  of  which  is  the  base  of  the  solid, 
and  the  other  is  a  plane,  parallel  with  the  plane  of  the  base,  pass- 
ing through  the  vertex  of  the  solid. 

17.  The  area  of  the  surface  is  measured  by  the  product  of  its 
length  and  breadth  (as  explained  by  scholium  on  page  32);  and 
these  dimensions  are  always  conceived  to  be  exactly  at  right 
i-ngles  with  each  other. 

18.  In  a  similar  manner,  solids  are  measured  by  the  product  of 
their  lengthy  breadth,  and  hight,  when  all  their  dimensions  are  at 
right  angles  with  each  other. 

The  product  of  the  length  and  breadth  of  a  solid,  is  the  measure 
of  the  iur/ac$  of  its  base. 


120  GEOMETRY. 

Let  P,  in  the  annexed  figure, 
represent  the  measuring  unit,  and 
AF  the  rectangular  solid  to  be 
measured. 

A  side  of  P,  is  one  unit  in 
length,  one  in  breadth,  and  one 
in  bight ;  one  inch,  one  foot,  one 
yard,  or  any  other  unit  that  may  be  taken. 

Then,         .         IXIX  1  =  1,  the  wwi^cttie. 

Now,  if  the  base  of  the  solid,  A  (7,  is,  as  here  represented,  6 
units  in  length  and  2  in  breadth,  then  it  is  obvious  that  (6x2= 10). 
10 units,  equal  to  P,  can  be  placed  on  the  base  of  AC,  and  no 
more ;  and  as  each  of  them  will  occupy  a  unit  of  altitude,  there- 
fore, 2  unUs  of  altitude  will  contain  20  solid  units,  3  units  of  alti- 
tude,  30  solid  units,  and  so  on ;  or,  in  general  terms,  the  number 
of  square  units  in  the  base,  multiplied  by  the  linear  units  in  perpendic- 
ttlar  altitude t  tuUl  give  the  solid  units  in  any  rectangular  solid.* 


THEOREM     1. 

Two  parallelopipedons  on  the  same  base,  and  of  the  same  altitude^ 
the  one  rectangular y  the  other  oblique^  the  opposite  sides  of  which  lie 
in  the  same  planes,  ipill  be  equal  in  solidity. 

Let  AG  he  the  rectangular  par- 
allelopipedon  on  the  base  A  C,  and 
AL  the  the  oblique  parallelopipedon, 
on  the  same  base,  AO,  and  of  the 
same  altitude,  namely,  the  perpen- 
dicular distance  between  the  par- 
allel planes  AC  and  JEL,  and  the 
side  AF,  in  the  same  plane  with  AIT,  and  the  side  DG,  in  the  same 
plane  with  DL.  Then  we  are  to  show,  that  the  oblique  parallelopip- 
edon ABCDMIKL,  is  equivalent  to  the  rectangular  parallelopip- 
edon, AG, 


*  This  is  one  of  those  simple  and  primary  truths  that  admit  of  no  demon- 
stration ;  for  no  other  truths  more  simple  and  elementary  than  itself  can  bo 
brought  to  bear  upon  it ;  hence  we  enunciate  it  as  a  definition. 

All  efforts  to  prove  a  proposition  which  is  perfectly  obvious,  are  very  unsat- 
isfactory to  the  mind,  and  always  tend  more  to  confuse  than  to  elucidate. 


BOOK    VII  121 

As  the  sides  of  the  two  solids  are  in  the  same  plane,  JEFK  is 
one  right  line  ;  EF=IKy  because  each  is  equal  to  AB,  From  the 
whole  line  EK,  subtract,  successively,  EF  and  IK;  thus  showing 
that  EI=^FK.  But  BF^AE,  and  the  angle  BFK=^  the  angle 
AEI;  therefore,  the  A  BFK^A  AEL  The  parallelogram  DE 
=  CFy  and  the  parallelogram  EM=FL;  and  all  the  angles  at  F 
forming  the  solid  angles  at  that  point,  are  respectively  equal  to  the 
like  angles  at  E. 

Hence,  the  two  prisms,  CBFGLK  2kTi6.  DAEHMI  qxq  equal;  for 
they  are  bounded  by  equal  planes  equally  inclined  to  each  other; 
or,  one  prism  can  be  conceived  to  be  taken  up  and  placed  into  the 
same  space  occupied  by  the  other;  and  magnitudes  that  fill  the 
same  space,  are  equal. 

Now,  from  the  whole  solid,  take  the  prism  GB — K,  and  the 
upright  solid,  A  0^  is  left ;  and  from  the  whole  soHd  take  the  prism 
DE — I,  and  the  oblique  solid,  AL^  is  left.  Hence,  by  (ax.  3)  the 
rectangular  parallelopipedon  AOy  is  equivalent  to  the  oblique 
parallelopipedon  AL^  on  the  same  base  and  altitude.     Q.  E,  D, 

C(yr.  The  measure  of  the  solid  ^4  6^,  is  the  base,  ABCD,  into  the 
perpendicular,  AE  (def.  18,  solid  ge.);  consequently,  the  measure 
of  the  solid,  AL,  is  also  the  same  base,  multiplied  by  the  same 
perpendicular. 

Scholium.  If  EF  and  IK  are  in  the  same  line ;  that  is,  the 
sides  AF  and  AK  in  the  same  plane  ;  but  the  angles  AEH  and 
BFG  not  right  angles,  then  neither  parallelopipedon  is  rectangular; 
but  they  are  proved  equal  in  exactly  the  same  manner;  that  is,  by 
proving  the  two  prisms  equal,  and  subtracting  each  in  succession 
from  the  whole  solid. 

Hence,  two  oblique  parallelopipedon^y  on  the  same  base,  and  of  the 
same  altititdey  whose  opposite  sid^s  are  between  the  same  planes,  are 
equal  in  solidity. 


Any  oblique  parallelopipedon  is  equivalent  to  a  rectangular  parallel' 
opipiedon  on  the  same  base  and  altitude. 


122  GEOMETRY 


Q       1'       H 


\rW^  /^ 


Let  AG  he  any  oblique  parallelopip- 
edon,  and  AL  a  rectangular  parallelo- 
pipedon,  on  the  same  base,  DB,  and 
between  the  same  parallel  planes,  BD 
and  HF.  Then  we  are  to  show,  thai  they 
are  equivalent. 

Produce  HO  and  IM;  and  because 
they  are  in  the  same  horizontal  plane,  and  not  parallel,  they  will 
meet  in  some  point,  Q,  Also  produce  FJS  and  KL,  and  thus  form 
the  parallelogram  iVP.  Now  conceive  another  parallelopipedon  to 
stand  on  the  base  DB,  and  its  upper  base  occupying  the  parallel- 
ogram NP=DB.  Now,  by  scholium  to  theorem  1,  book  7,  the 
solid,  AOy  is  equal  to  this  imaginary  solid,  AP.  But  (th.  1,  b.  7), 
the  rectangular  solid,  AL,  is  also  equal  to  this  imaginai-y  solid, 
AP.  Therefore,  the  solid  -4  (r  is  =  to  the  rectangular  solid,  AL. 
(Ax).     Q.E.D. 

Cor.  Hence,  every  parallelopipedon,  in  whatever  direction  or  degree 
it  is  incliyied,  is  measured  hy  the  product  of  its  base  into  its  perpen- 
dicular altittide. 

THE  O  REM    3. 

ParaUelopipedons  on  the  same,  or  on  equal  bases,  are  to  one  another 
as  their  perpendicidar  altitudes;  and  parallelopipedons  having  equal 
cdtitudes,  are  to  one  another  as  their  bases. 

Let  P  and  p  represent  two  parallelopipedons,  whose  bases  are  B 
and  b,  and  altitudes  A  and  a. 

Then,  by  the  last  theorem,  the  measure  of  P  is  BA,  and  the 
measure  of  p  is  ba.  But,  all  magnitudes  are  proportional  to  their 
numerical  measures  ;  that  is,  .         .         ,  P  :  p=BA  :  ba 

Now,  in  case  -4=a,  we  have  (th.  4,  b.  2),       P  :  p-^B  :  b 
In  case  B=b,  then  we  have,         .         .         .  P  :  p=A  :  a 

Q.  E.  D. 

THEOREM     4. 

Similar  parallelopipedons  are  to  one  another  as  the  cubes  of  their 
like  dimensions.* 

*  This  theorem  Is  true  for  all  similar  solids. 


BOOK    VII  128 

Let  P  and  p  represent  two  parallelopipedons,  as  in  theorem  3; 
and  let  I  and  n  represent  the  length  and  breadth  of  the  base  of 
Py  and  h  its  altitude. 

Also,  let  V  and  n'  represent  the  length  and  breadth  of  p,  and  h! 
its  altitude. 

Hence,  by  cor.  to  th.  2,  b.  7,  P=lnh,  and^=ZVA'. 

That  is,     .        .        P  :  p=lnli  :  I'n'h"*' 

But,  by  reason  of  the  similarity  of  the  solids, 
/  :  l'=:n  :  n' 
n  :  n'=n  :  n' 

And,        •        .         k :  h'=n  :  n' 

Multiplying  these  proportions  together,  term  by  term,  (th.  11  b.2), 
we  have,      .        .  Ink  :  l'n'h'=^v?  :  n'* 

That  is,    .         -         P  \p=n^  :  n'^         (th.  6,  b.  2) 

By  a  little  different  arrangement  of  the  proportions, 
we  have,  P  :  p=P  :  l'^ 

Or,  ,         .         .         P  :p=h':h'*  Q.  E.D. 

THEOREM    5. 

Any  parallelopipedon  may  he  divided  into  two  equal  prisms ^  by  a 
diagonal  plane  passing  through  its  opposite  edges. 

The  parallelopipedon  may  be  conceived 
to  be  composed  of  a  great  multitude  of  ex- 
tremely thin  parallelograms,  all  equal  to  one 
another;  and  the  diagonal  HF  divides  the 
parallelogram  EG  into  two  equal  parts  (th. 
22,  cor.  b.  1 ) ;  and  the  line  IIF,  passing  down 
through  all  the  parallelograms,  from  EO  to 
A  (7,  divides  each  and  all  of  them  into  two  equal  parts ;  that  is, 
the  diagonal  plane,  HFBD,  divides  the  parallelopipedon  into  two 
equal  parts,  each  of  which  is  a  prism.     Q.  E.  D. 

Otherwise,  the  two  prisms  may  be  proved  to  be  bounded  by 
equal  planes  and  equal  angles  ;  therefore,  they  are  magnitudes  that 
exactly  fill  equal  spaces,  and  are  therefore  equal.     Q.  E.  D. 

*  When  the  three  factors  are  all  equal ;  that  is,  Z=n=^,  P  :  p=P  :  fa  j 
but  in  this  case,  the  solids  are  actual  cubes. 


124  GEOMETRY. 

Cor,  The  solidity  of  a  prism  is  therefore  the  triangular  base, 
DBGt  multiplied  by  its  altitude,  the  perpendicular  distance  between 
the  planes  A  C  and  EO;  or,  it  may  be  found  by  the  product  of  the 
base,  HQCDf  and  half  the  perpendicular  distance  between  the 
planes  QD  and  EB, 

THEOREM    6. 

All  prisms  of  equal  hoses  and  altitudes  are  equal  in  solidity,  what- 
ever he  the  figures  of  the  bases. 

It  is  of  no  consequence  what  shape  a  base  may  be,  for  it  is 
greater  or  less,  according  to  the  number  of  square  units  that  may 
be  contained  in  it ;  hence,  the  base  of  a  triangular  prism  may  be 
considered  a  square,  or  rectangular  prism,  containing  the  same 
number  of  square  units  as  the  triangular  base  ;  that  is,  any  prism 
may  be  considered  a  rectangular  parallelopipedon,  whose  base  is 
the  same  in  area  as  the  base  of  the  prism ;  but  the  solidity  of  a 
parallelopipedon  is  measured  by  the  area  of  its  base  by  its  altitude 
(def.  18)  ;  and  therefore,  a  prism  of  the  same  area  of  base  and 
altitude,  has  the  same  measure.     Q.  E.  D» 


THEOREM    7. 

AU  similar  solids  are  to  one  another  as  the  cubes  of  their  like 
dimensions. 

By  theorem  4,  of  this  book,  this  proposition  is  proved  true 
for  all  similar  parallelopipedons ;  and  by  theorem  6,  all  similar 
parallelopipedons  may  be  divided  into  two  equal  parts,  thus 
forming  similar  prisms.  But  the  halves  of  things  are  in  the 
same  proportion  as  their  wholes  ;  therefore,  all  similar  prisms  are 
to  one  another  as  the  cubes  of  their  like  dimensions. 

Similar  pyramids  and  similar  cones  are  but  the  same  like  parts 
of  similar  prisms ;  and,  like  parts  of  wholes,  are  in  the  same  pro- 
portion as  the  wholes  themselves ;  therefore,  our  theorem  is  true 
for  pyramids  and  cones. 

Spheres  are  like  proportional  parts  of  their  circumscribing  cyl- 
inders ;  and  our  theorem  is  true  for  similar  cylinders  ;  it  is,  there- 
fore, true  for  spheres. 


BOOK    VII 


125 


In  short,  all  similar  solids,  however  irregular'  the  shape,  are  but 
like  parts  of  some  mathematical  figure  that  may  inclose  them  ;  and 
as  the  theorem  is  true  for  the  mathematical  figures,  it  is  true  for 
any  of  their  like  parts ;  it  is,  therefore,  true  for  all  similar  solids 
whatever.     Q.  E.  D. 

TH  EOREM     8. 

If  a  pyramid  be  cut  by  a  plane  which  is  parallel  ivith  its  base,  the 
section  thus  formed  will  be  similar  to  the  base,  and  its  area  will  be  to 
the  area  of  the  base  as  the  square  of  its  perj^endicular  distance  from 
the  vertex,  is  to  the  square  of  the  perpendicular  altitvde  of  the  pyramid. 

Let  MN  and  mri  be  two  par- 
allel planes,  between  which 
stands  any  pyramid  whose 
base  is  P,  and  vertex  O,  and 
perpendicular  altitude  EF. 

On  any  one  of  the  edges,  as 
OAy  take  any  point  a,  and 
draw  ab  parallel  to  AB  ;  and 
from  h  draw  be  parallel  to  BQ.  Then,  by  reason  of  the  parallels 
(th.  10,  b.  1),  the  angle  abc=zABC.  In  this  manner  we  may  go 
round  the  whole  section,  whatever  be  the  number  of  sides :  and 
every  angle  in  the  section  will  be  equal  to  its  corresponding  angle  of 
the  base  ;  that  is,  the  two  figures  are  equiangular,  and  similar ;  and 
as  every  line  of  the  section  is  parallel  to  its  corresponding  line  in  the 
base,  therefore,  if  the  base  is  a  plane,  the  section  will  be  a  parallel 
plane.     Produce  a  line  from  this  plane  to  the  perpendicular  at  H. 

But  equiangular  plane  figures  are  to  one  another  as  the  squares 
of  their  like  sides  (th.  23,  b.  2);  that  is, 
P  :  p=AB'  :  ab^ 

But,       .      AB'  :  (aby=GA'  :  Ga^     (th's.  17  and  10,  b.  2) 

And,      .       OA^  :     Ga^^GE' :  Ge' 

And,     .      GE^  :     Ge^=FE^  :  FII^ 

Multiplying  all  these  proportions  together,  and  at  the  same  time 
rejecting  all  the  common  factors  that  would  otherwise  appear  in 
the  antecedents  and  consequents,  we  have, 
P:p=^FE^:FB^ 


126  GEOMETRY 

By  changing  means  for  extremes,  we  have, 

jp  :  P=FH^ ;  FE^  §.  E.  D, 

C(yr.  As  the  section  made  by  the  cutting  plane  is  always  similar 
to  the  base,  it  follows  that  when  the  base  is  a  polygon  of  a  great 
number  of  sides,  the  section  will  be  a  polygon  of  the  same  number 
of  sides ;  and  when  the  base  is  a  circle,  the  section  will  be  a 
circle,  and  so  on. 

THEOREM    9. 

If  two  pyramids,  standing  between  two  parallel  planes,  he  cut  by  a  third 
parallel  plane,  (he  respective  sections  will  be  to  each  other  as  their  bases. 

Let  two  pyramids  stand  as 


JiltziM 


^/  \n 


represented  in  the  figure,  and 
from  any  point,  If,  in  the  per- 
pendicular, pass  a  plane  par- 
allel to  the  plane  JfiV.  By 
the  last  theorem,  each  sec- 
tion of  these  pyramids  is  a 
similar  figure  to  its  base. 

By  theorem  6,  book  6,  the  parallel  plane  that  forms  these  sec- 
tions, cuts  all  lines  between  the  planes  MJ^smdmn,  proportionally. 

Therefore,        .         gr  :  gIi=Qe  :  GE 

And,        .        .         Ge  :  GE=EJI:  FE 

Hence,     .        .  gr  :  gR=FII :  FE 

By  squaring  this  last  proportion,  we  have, 
gr" :  gK'^FH^  :  FE^ 

But,  .         .         gr":  gR^=rs^  :  JIS^ 

By  the  application  of  theorem  6,  book  2,  to  these  last  two  pro- 
portions, we  have,    Fff^  :  FE^=rs^  :  BS^ 

But,         .         .         .     p:  P=FIP-  :  FE^  (th.  8,  b.  7) 

And,        .         .        rs^:  IiS'=q  :  Q  (th.  23.  b.  2) 

Multiplying  these  three  proportions  together,  term  by  term,  re- 
jecting common  factors  in  antecedents  and  consequents,  we  have, 
p  :  F=g  :  Q  Q.  E.  D. 

Cor.   On  the  supposition  that         P=Q,  there  results    p=q. 

THEOREM     10. 

Any  two  pyramids  haxing  equal  bases,  and  sittiated  between  the  sam£ 
two  parallel  planes f  or  having  eqical  altitudes ,  are  equal. 


BOOK    VII  127 

Take  the  same  figure  as  for  the  last  theorem,  supposing  the 
bases,  P  and  §,  equal,  and  conceive  the  perpendicular  EF^  to  be 
divided  by  a  great  multitude  of  parallel  planes,  equidistant  from 
each  other,  and  all  parallel  to  the  plane  MN.  By  the  last  theorem, 
these  planes  will  divide  each  pyramid  into  the  same  number  of 
equal  parallel  sections,  of  which  the  two  pyramids  may  be  con- 
sidered as  composed  ;  and,  as  the  sums  of  equals  are  equal,  there- 
fore, the  two  pyramids  are  equal.     Q,  E.  D, 

THEOREM     11. 

Every  triangular  pyramid  is  a  third  pari  of  the  triangvlar  prism, 
having  the  same  base  and  the  same  altitude. 

Let  FABG  be  a  triangular  pyramid; 
ABCDEF  a  triangular  prism  of  the  same 
base  and  the  same  altitude :  the  pyramid 
will  be  equal  to  a  third  of  the  prism. 

Cut  off  the  pyramid  FABQ  from  the 
prism,  by  a  section  made  along  the  plane 
FA  C;  there  will  remain  the  solid  FA  ODE, 
which  may  be  considered  as  a  quadrangular 
pyramid,  whose  vertex  is  F,  and  whose  base  is  the  parallelogram 
A  ODE.  Draw  the  diagonal  CE;  and  extend  the  plane  FCEy 
which  will  cut  the  quadrangular  pyramid  into  two  triangular  ones, 
FA  CEj  FCDE.  These  two  triangular  pyramids  have  for  their 
common  altitude,  the  perpendicular  let  fall  from  F  on  the  plane 
ACDE.  They  have  equal  bases,  the  triangles  ACE,  QBE, 
being  halves  of  the  same  parallelogram  ;  hence,  the  two  pyramids, 
FACE,  FCDE,  are  equivalent  (th.  10,  b.  7).  But  the  pyramid 
FCDE,  and  the  pyramid  FABC,  have  equal  bases,  ABC,  DEF; 
they  have,  also,  the  same  altitude,  namely,  the  distance  of  the 
parallel  planes  ABC,  DEF;  hence  these  two  pyramids  are  equi- 
valent. Now,  the  pyramid  FCDE  has  already  been  proved  equi- 
valent to  FACE;  consequently,  the  three  pyramids,  FABC, 
FCDE,  FA  CE,  which  compose  the  prism  ABD,  are  all  equivalent. 
Hence,  the  pyramid,  FABC  is  the  third  part  of  the  prism  ABD, 
which  has   the  same  base,  and  the    same  altitude.      Q.  E.  D. 

Cor,  The  solidity  of  a  triangular  pyramid  is  equal  to  a  third 
part  of  the  product  of  its  base  by  its  altitude. 


128  GEOMETRY 

The  preceding  demonstration  is  brief,  direct,  and  all  that  could 
be  desired,  provided  the  learner  has  a  clear  coAception  of  the 
figure  as  represented  on  paper ;  but  as  we  know  that  this  is  not 
generally  the  case,  we  give  the  following  method. 

Let  ABCDEF  be  any  rectangular  par- 
allel opipedon,  and  put  AD=ay  AB=bf  and 
AF=h.  Produce  AF  to  Oy  making  FO 
=AF.  Draw  00  to  meet  AB,  produced 
in  M.  As  FO  is  parallel  to  AB,  and  AG 
double  of  AF,  therefore,  AM  is  double  of 
AB.  Join  6^-£','and  produce  it  to  meet  AD, 
in  I;  then,  by  like  reasoning,  we  shall  find  AI  the  double  of  AD, 
Join  Gil,  and  produce  it  to  meet  the  plane  of  BD,  in  Q. 

The  whole  figure  now  comprises  two  pyramids ;  one,  whose  base 
is  A  Q;  the  other  similar  one  has  FIT  for  its  base,  and  the  vertex 
of  both,  is  G. 

The  whole  figure  also  comprises  the  parallelopipedon  AJI,  which 
is  measured  by  (abk),  two  prisms,  and  two  equal  and  similar  pyra- 
mids. One  prism  has  DCKIiov  its  base,  and  DE,  for  its  altitude  ; 
the  other  has  BMLQ  for  its  base,  and  BO—DE,  for  its  altitude. 

As  each  of  these  bases,  i>j5^  and  BL,  is  equal  to  AC,  hence, 
the  solidity  of  these  two  prisms  is  expressed  by  (abh);  and  the 
parallelopipedon,  and  two  prisms  together,  are  measured  by  2abk  ; 
and,  in  addition  to  these,  we  have  two  equal  pyramids  of  unknovm 
solidity;  therefore,  let  each  one  be  represented  by  x. 

Now,  the  whole  pyramid,  whose  base  is  AQ,  and  vertex  G,  is 
expressed  by  (2a5/i-|-2ar). 

But  the  pyramid,  whose  base  is  FH,  and  vertex  G,  is  expressed 
by(x). 

As  these  two  pyramids  are  similar,  they  are  to  each  other  as  the 
cubes  of  their  like  dimensions ;  that  is,  they  are  to  each  other  as 
the  cube  of  GA  to  the  cube  of  GF.  But  GA  is  the  double  of 
OF,  by  construction.     Therefore,  OA^  :  OF^=S  :  1 

Hence,     ....  (2abk-{-2x)  :  x=8  :  1 

Product  of  extremes  and  means  gives,    8x=2abh-{'2x 

Therefore, x=^(abh) 

This  last  equation  shows  that  the  solidity  of  any  pyramid  is  me- 
third  of  any  rectangular  solid  of  the  same  base  and  altitude. 


BOOK   VII.  129 

Cor,  This  measure  of  the  pyramid  is  true,  whatever  be  the 
figure  of  its  base ;  and  when  the  base  is  a  circle,  the  pyramid  is 
called  a  cone  ;  hence,  the  solidity  of  a  cone  is  one  third  of  its  cir- 
cumscribing cylinder. 

THEOREM    12. 

If  a  pyramid  he  cut  hy  a  plane  parallel  to  Us  base,  the  solidity  of 
the  frustum  that  remains  after  the  small  pyramid  is  taken  away,  is 
equal  to  three  pyramids  of  the  same  altitude  as  the  frustum;  one  hav- 
ing for  its  base,  the  hose  of  the  frustum;  another,  the  upper  base;  and 
the  third,  a  base  which  is  the  mean  proportional  between  the  upper  and 
lower  bases  of  the  frustum. 

(The  figure  has  been  previously  described  in  theorem  8.) 

Now,  by  the  last  theorem,  the  solidity  of  the  whole  pyramid  is 

P(  WF^\  n(  JPF1  \ 

expressed  by  — ^^ — ^,  and  that  of  the  small  pyramid  is  ^       - 

The  difference  of  these  magnitudes  measures  the  frustum ; 

That  is,         .         .      -^ ^~^ ^=the  frustum. 

To  make  this  expression  cor- 
respond with  the  enumeration 
of  this  theorem,  we  must  ban- 
ish FE  and  FH,  and  obtain 
their  difference. 

By  th.  8,  book  7,  we  have, 

FF:FE=JP:J~p     (1) 

From  this    proportion  we 

have,  FF=^ ^— ,  which,  substituted  in  the  above  expression, 

(FH)PJp    p(FH)        ^    , 

gives,    .      ^ i-J±L,^\       f=  the  frustum  ; 

Zjp        ^    3    ^ 

Or,       .      (^^)t^/-^^>ZL)=  the  frustum. 

From  proportion  (1),  FE—FH  :  FH=JP—Jp  :  ^p     (2) 
But  {FE—FH)  is  the  altitude  of  the  frustum,  which  we  will 
designate  by  a. 

Then,  from  proportion  (2),  FH=i    ^^ — 
9  JP-JP 


130  GEOMETRY 

This  value  of  FH^  substituted  in  the  last  expression  for  the 
frustum,   gives, 

e(^^^lf#)  =  the  frustum. 
By  actual  division,  we  have, 

\(P-\-  JPl>+p)==  the  frustum  ; 

Or,         .      iaP-\-^aJFp-^iap=  the  frustum. 

Here  we  find  expressions  for  three  different  pyramids,  which, 
together,  are  equal  to  the  frustum ;  one  has  P  for  its  base,  another 
p,  and  the  third  JPp,  which  is  the  mean  proportional  between  the 
two  bases,  P  andj9;  therefore,  a  frustum  is  equal,  <fec.     Q.  E.  D. 

Cor.  In  case  P=p,  the  frustum  becomes  a  prism,  and  the  above 
expression  for  the  three  pyramids  becomes  aP,  which  is  tlie  proper 
expression  for  the  solidity  of  a  prism. 

THEOREM     13. 

The  co7ivex  surface  of  any  regtdar  pyramid  is  equal  to  the  perimeter 
of  its  hasBj  mtdtipled  by  half  its  slant  highi. 

Bisect  the  side  AB  in  H,  and  join  SE, 
Since  the  pyramid  is  regular,  the  side  SAB 
is  an  isosceles  triangle ;  consequently,  SH 
is  perpendicular  to  AB;  hence,  SH  is  the 
altitude  of  the  triangle,  and  also  the  slant 
bight  of  the  pyramid.  For  the  same  reason, 
each  side  of  the  pyramid  is  an  isosceles  tri- 
angle, whose  altitude  is  the  slant  bight  of 
the  pyramid. 

Now,  the  area  of  tlie  triangle  SAB,  is 
equal  to  ABX^SH;  and  the  area  of  all  the  triangles  which 
compose  the  convex  surface  of  the  pyramid,  is  equal  to  the  sum  of 
their  bases.       {AB-\-BC-\- CD-['DE-\-EF-\-AF)X \SK 

But  the  sum  of  these  bases,  AB,  BC,  <fec.,  forms  the  perimt-ter 
of  the  pyramid's  base  ;  and  the  common  altitude,  SH,  is  tlie  sh\nt 
hight  of  the  pyramid.  Therefore,  the  convex  surface  of  any  regular 
pyramid,  is  equal  to  the  perimeter  of  its  base  multiplied  by  half  its 
slant  hiyht. 


BOOK    VII. 


131 


„/  if  \.. 

w 

> 

r 

1 

n^ 

H 

jKi^ 

THEOREM    14. 

jTAc  convex  surface  of  a  frustum  of  a  regular  pyramid,  is  equal  to 
the  sum  of  the  perimeter  of  the  two  bases  multiplied  by  half  the  slant 
hight. 

Conceive  a  regular  frustum  of  a  pyramid  to 
exist,  as  represented  in  the  figure ;  then  each 
face  will  be  a  regular  trapezoid,  whose  surface 
is  measured  by  the  half  sum  of  its  parallel 
sides  (th.  31,  b.  1),  multiphed  by  the  perpen- 
dicular distance  between  them,  which  is  the 
slant  hight  of  the  frustum. 

Let  S  represent  a  side  of  the  lower  base, 
and  s  a  side  of  the  upper  base,  and  a  the  slant 
hight ;  then  the  surface  of  one  face  is  measured 
by  ia  (S+^). 

There  are  just  as  many  of  these  surfaces  as  the  frustum  has 
sides.  Let  m  represent  the  number  of  sides  ;  then  the  whole  sur- 
face must  be  ^a{mS-\-7ns).  But  [mS-\-ms),  is  the  perimeter  of 
the  two  bases  ;  and  ^a  is  one-half  of  the  slant  hight.  Therefore, 
&c.     Q.  E.  D. 

Scholium.  Let  circles  be  described  round  the  bases  of  the 
frustum,  as  represented  in  the  last  figure  ;  and  conceive  the  number 
of  sides  to  be  indefinitely  increased ;  then  S  and  s  will  be  indefi- 
nitely small,  and  m  indefinitely  great ;  but  however  small  S  and 
«  may  be  (the  corresponding  number  to  m  being  as  much  in- 
creased), the  expression  [mS-{-ms)  will  still  represent  the  perime- 
ters of  the  two  bases.  But,  when  S  and  s  are  indefinitely  small, 
while  OA,  and  DII,  that  is,  the  distances  from  the  axis  of  the 
frustum  from  its  edges  being  constant,  the  perimeter,  mS,  wil- 
become  the  perimeter  of  the  circle  of  which  OA  is  the  radius; 
and  ms  will  be  the  perimeter  of  the  circle  of  which  DII  is  the  ra- 
dius  ;  that  is,  mS=^n{A  0),  and  ms=2ft(DB);  and  by  addition. 
mS-{-ms=27t{AO-^J)H) 

But,  in  this  case,  ^  becomes  -J^^i),. one-half  the  edge  of  the 
frustum  ;  and  the  frustum  of  the  pyramid  becomes  the  frustum  of 
ft  cone,  and  its  surface  is  measured  by 

^ADX  2rt(A  0-\-DH);  hence, 


132  GEOMETRY. 

7^  convex  surface  of  a  frustum  of  a  cone,  is  equal  to  half  its 
sides,  multipled  hy  the  sum  of  the  circumferences  of  its  two  bases. 
The  above  expression  is  the  same  as 

If  we  take  the  middle  point,  P,  between  0  and  E,  and  draw 
PM  parallel  to  OA  and  ED, 

Then,        .        .        . =^PM,  which,  substituted, 

gives     ....  ADX^TtPM 

That  is,  the  convex  surface  of  the  frustum  of  a  cone,  is  equal  to  its 
side,  multiplied  hy  the  circumference  of  a  circle  which  is  exactly  midway 
between  its  two  bases, 

THEOREM    15. 

Jf  any  regular  semi-polygon  be  revolved  abotd  its  axis,  the  surface 
thus  described,  will  be  measured  by  the  product  of  its  axis  into  the  cir- 
cumference of  its  inscribed  circle. 

If  the  semi-polygon,  DABK,  <fec.,  revolve 
on  its  axis,  DE,  the  sides  AB,  BK,  &c.,  will 
each  describe  frustums  of  cones ;  and,  for  in- 
vestigation, let  us  take  the  side  AB, 

From  the  middle  point,  O,  draw  GI  perpen- 
dicular to  DE,  Join  OC,  and  draw  -4  2^  parallel 
toi>^. 

By  the  scholium  to  the  preceding  theorem, 
the  surface  described  by  AB  is  measured  by 
ABX   cir.  GI,  which  is  equal  to  AT,  or  EL 

dr. GO, 

That  is,        .        ELX^TtGC=ABx^^Gl 

The  two  triangles,  ABT  and  CGI,  are  similar.  As  CG  is  per- 
dendicular  to  AB,  the  two  angles  CGI  and  IGA,  are  equal  to  a 
right  angle.  The  acute  angles  of  the  A  ABT  are  also  equal  to  a 
right  angle. 

That  is,        .  jCGI-\-aIGA=jBAT-^aABT 

But,     .        .        .  JIGA=  J  ABT  {th,5,h,l) 

By  subtraction,         .        J  CGI=  J  BAT 


BOOK    VII.  133 

Now,  as  these  two  triangles  have  each  a  right  angle,  they  are 
equiangular  and  similar; 

Therefore,       .     CG:  01=: AB  :  AT^ffL 

Hence,   .        .      HL-CG^AB^QI 

Multiplymg  both  members  of  this  equation  by  2;*,  we  have, 
HL-^tt  CGh=^AB'^7t  01 

Thus  we  find  that  the  surface  described  by  the  side  ABy  is  mea- 
sured by  the  product  of  EL  into  the  circumference  of  the  inscribed 
circle ;  and  in  the  same  manner  we  may  prove  that  the  surface 
described  by  the  side  AD,  is  measured  by  BE  into  the  circum- 
ference of  the  same  circle,  and  so  on  of  every  other  side  ;  and  the 
surface  described  by  all  the  sides  taken  together,  is  equal  to 
(DE'-{-EL'\-ZC,  (fee),  multiplied  into  the  circumference  of  the 
inscribed  circle ;  that  is,  the  surface  described  by  the  whole  poly- 
gon, is  equal  to  EU,  the  axis  of  the  polygon,  into  the  circumference 
of  its  inscribed  circle.     Q,  E.  E. 

THEOREM    16. 

The  convex  surface  of  a  sphere  is  equal  to  the  ^product  of  its  dia- 
meter into  its  circumference. 

The  last  theorem  is  true,  whatever  be  the  number  of  sides  of 
the  polygon ;  and  now  suppose  the  number  to  be  indefinitely  great ; 
then  the  sides  of  the  polygon  will  coincide  with  the  circumference 
of  the  circle,  and  CO  becomes  CA,  and  the  surface  described  by 
the  sides  of  the  polygon,  is  now  the  surface  of  the  sphere,  which 
is  measured  by  the  diameter  EE,  multiplied  into  the  circumference 
of  the  circle  2nCA.     Q.  E.  E, 

Cor.  1 .  If  we  represent  the  radius  of  a  sphere  by  R,  its  circum- 
ference is  2rti2,  and  its  diameter  2i2;  therefore,  its  convex  surface 
is  AhB?.  The  surface  of  a  plane  circle,  whose  radius  is  i?,  is  hR^; 
tlierefore,  the  surface  of  a  sphere  is  4  times  a  plane  circle  of  the  same 
diameter. 

Cor.  2.  The  surface  of  a  segment  is  equal  to  the  circumference 
|Hj|     of  the  sphere,  multiplied  by  the  thickness  of  the  segment. 

Cor.  3.  In  the  same  sphere,  or  in  equal  spheres,  the  surfaces  of 
different  segments  are  to  each  other  as  their  altitudes. 


134  GEOMETRY 


THEOREM     17 


The  solidity  of  a  sphere  is  equal  to  the  product  of  tts  surface  into 
a  third  of  its  radius. 

Let  us  suppose  a  sphere  to  be  composed  of  a  great  multitude  of 
regular  pyramids,  whose  bases  are  portions  of  the  surface  of  the 
sphere,  and  their  common  vertex  the  center  of  the  sphere  ;  then  the 
altitudes  of  all  such  pyramids  is  the  radius  of  the  sphere. 

The  solidity  of  one  of  these  pyramids  is  its  base  multiplied  by 
•J  of  its  altitude  (th.  11,  b.  7);  and  the  solidity  of  all  of  these 
together,  will  be  the  sum  of  all  the  bases  multiplied  into  ^  of  the 
common  altitude.  But  the  sum  of  all  the  bases,  is  the  surface  of 
the  sphere  ;  and  the  common  altitude  is  the  radius  of  the  sphere  ; 
therefore,  the  solidity  of  a  sphere  is  equal  to  its  surface  multiplied 
by  one  third  of  its  radius.     Q.  E.  D. 

Let  R  =  the  radius  of  the  sphere  ;  then  (cor.  1,  th.  16,  b.  7), 
^TiB}  is  its  surface ;  hence,  its  solidity  must  be 

Cor.  If  r  represent  the  radius  of  any  other  sphere,  its  solidity 
will  be  l^r* ;  and,  by  dividing  by  the  constant  factors,  ^n,  these 
two  solids  are  to  each  other  as  jR'  to  r*,  a  result  corresponding  to 
theorem  7,  book  7. 

THEOREM     18. 

The  solidity  of  a  sphere  is  two-thirds  the  solidity  of  its  circumscrib- 
ing cylinder. 

Let  i2  be  the  radius  of  the  base  of  an  upright  cylinder ;  then, 
hB^  will  be  the  area  of  the  base  (th.  1,  b.  6);  but  the  altitude  of 
a  cylinder  which  will  just  inclose  a  sphere,  must  be  2i2;  and  the 
solidity  of  such  a  cylinder  must  be  ^nl^  (def.  18,  b.  7).  By  the 
last  theorem,  the  solidity  of  a  sphere,  whose  radius  is  It,  is  |rti?. 

Therefore,  the  cylinder  is  to  the  sphere  as    '^hR^  to  ^kR^ 

Or,  as 2         to  f 

Or,  as 1         to  f 

Q.  E,  i>. 


BOOK     VII. 


135 


We  ^ve  another  method  of  demonstrating  this  truth,  merely  for 
the  beauty  of  the  demonstration. 

Let  AK  he  the  diameter  of  a  semicircle,  and 
also  the  side  of  a  parallelogram  whose  width  is 
the  radius  of  the  semicircle.  M^KSHHSI 

Join  the  center  of  the  semicircle  to  either  ex- 
tremity of  the  parallelogram,  as  CJB,  CL.  Now  Qg^^^^SJ 
conceive  the  parallelogram  to  revolve  on  AK, 
and  it  will  describe  a  cylinder;  the  semicircle 
will  describe  a  sphere,  and  the  triangle  ABQ 
will  describe  a  cone. 

In  A  Cy  take  any  point,  2),  and  draw  DJU  par- 
allel to  AB,  and  join  CO.     Then,  as  CA=AB,  CD=DB.    In 
the  right  angled  triangle  CD  0,  we  have, 

CD^-\-DO^=CO^    (1) 
But,        .        .        .         BD^=:DE\  and  CO^^DH^ 
Substituting  these  values  in  equation  (1),  and  we  have, 

CE^'\'DO^=DH^    (2) 
Multiply  every  term  of  this  equation  by  n. 
Then,      .        .  TtDE  ^-\-rtD  O^^^nDH^ 

Now,  the  first  term  of  this  equation,  is  the  measure  of  the  sur- 
face of  a  plane  circle,  whose  radius  is  DE;  the  second  term  is  the 
measure  of  a  plane  circle,  whose  radius  is  D  0;  and  the  second 
member  is  the  measure  of  the  surface  of  a  plane  circle,  whose  radius 
is  DH.  Let  each  of  these  surfaces  be  conceived  to  be  of  the  same 
extremely  minute  thickness  ;  then  the  first  term  is  a  section  of  a  cone, 
the  second  term  is  a  corresponding  section  of  a  sphere,  and  these  two 
sections  are,  together,  equal  to  the  corresponding  section  of  the 
cylinder;  and  this  is  true  for  all  sections  parallel  to  CR,  which 
compose  the  cone,  the  sphere,  and  the  cylinder  ;  therefore,  the 
cone  and  sphere,  together,  are  equal  to  the  cylinder ;  but  the  col^ 
described  by  the  triangle  ABC,  is  \  of  the  cylinder  described  by 
AR  (th.  11,  b.  7);  therefore,  the  corresponding  section  of  the 
sphere,  is  the  remaining  tioo-thirdsy  and  the  whole  sphere  is  two- 
thirds  of  the  whole  cylinder  described  by  the  parallelogram  AL. 

Q,  E.  i>. 


136  ELEMENTS    OF 


ELEMENTARY    PRINCIPLES    OF    PLANE 
TRIGONOMETRY. 

Trigonometrt  in  its  literal  and  restricted  sense,  has  for  its  object, 
tlie  measure  of  triangles.  When  the  triangles  are  on  planes,  it  is 
plane  trigonometry,  and  when  the  triangles  are  on,  or  conceived  to 
be  portions  of  a  sphere,  it  is  spherical  trigonometry.  In  a  more 
enlarged  sense,  however,  this  science  is  the  application  of  the  prin- 
ciples of  geometry,  and  numerically  connects  one  part  of  a  magni- 
tude with  another,  or  numerically  compares  dififerent  magnitudes. 

As  the  sides  and  angles  of  triangles  are  quantities  of  different 
kinds,  they  cannot  be  compared  with  each  other ;  but  the  relation 
may  be  discovered  by  means  of  other  complete  triangles,  to  which 
the  triangle  under  investigation  can  be  compared. 

Such  other  triangles  are  numerically  expressed  in  Table  II,  and 
all  of  them  are  conceived  to  have  one  common  point,  the  center  of 
a  circle,  and  as  all  possible  angles  can  be  formed  by  two  straight 
lines  drawn  from  the  center  of  a  circle,  no  angle  of  a  triangle  can 
exist  whose  measure  cannot  be  found  in  the  table  of  trigonometrical 
lines. 

The  measure  of  an  angle  is  the  arc  of  a  circle,  intercepted  be- 
tween the  two  lines  which  form  the  angle — the  center  of  the  arc 
always  being  at  the  point  where  the  two  lines  meet. 

The  arc  is  measured  by  degrees^  mimttes,  and  seconds,  there  being 
360  degrees  to  the  whole  circle,  60  minutes  in  one  degree,  and  60 
seconds  in  one  minute.  Degrees,  minutes,  and  seconds,  are  desig- 
nated by  °,  ',  ".  Thus  27°  14'  21",  is  read  27  degrees,  14  min- 
utes, and  21  seconds. 

All  circles  contain  the  same  number  of  degrees,  but  the  greater 
the  radii  the  greater  is  the  absolute  length  of  a  degree  ;  the  cir- 
cumference of  a  carriage  wheel,  the  circumference  of  the  earth,  or 
the  still  greater  and  indefinite  circumference  of  the  heavens,  have 
the  same  number  of  degrees ;  yet  the  same  number  of  degrees  in 
each  and  everv  circle  is  precisely  the  same  angle  in  amount  or 
measure. 


PLANE    TRIGONOMETRY.  137 

As  triangles  do  not  contain  circles,  we  can  not  measure  triangles 
by  circular  arcs  ;  we  must  measure  them  by  other  trmngles,  that  is, 
by  straight  lines,  drawn  in  and  about  a  circle,    from  the  center. 

Such  straight  lines  are  called  trigonometrical  lines,  and  take  par- 
ticular names,  as  described  by  the  following 

DEFINITIONS. 

1.  The  sine  of  an  angle,  or  an  arc,  is  a  line  drawn  from  one  end 
of  an  arc,  perpendicular  to  a  diameter  drawn  through  the  other  end. 
Thus,  JBF  is  the  sine  of  the  arc  AB,  and  also  of  the  arc  BJDK  BK 
is  the  sine  of  the  arc  BD,  it  is  also  the  cosine  of  the  arc  AB,  and 
BFy  is  the  cosine  of  the  arc  BD. 

N.  B.  The  complement  of  an  arc  is  what  it 
wants  of  90° ;  the  supplement  of  an  arc  is 
what  it  what  it  wants  of  180°. 

2.  The  cosine  of  an  arc  is  the  perpendicu- 
lar distance  from  the  center  of  the  circle  to 
the  sine  of  the  arc,  or  it  is  the  same  in  mag- 
nitude as  the  sine  of  the  complement  of  the 
arc.  Thus,  CF,  is  the  cosine  of  the  arc  AB;  but  CF=KBt  the 
sine  of  BD, 

3.  The  tangent  of  an  arc  is  a  line  touching  the  circle  in  one 
extremity  of  the  arc,  continued  from  thence,  to  meet  a  line  drawn 
through  the  center  and  the  other  extremity. 

Thus,  AH  is  the  tangent  to  the  arc  AB,  and  DL  is  the  tangent 
of  the  arc  DB,  or  the  cotangent  of  the  arc  AB. 

N.  B.   The  CO,  is  but  a  contraction  of  the  word  complcTnent. 

4.  The  secant  of  an  arc,  is  a  line  drawn  from  the  center  of  the 
circle  to  the  extremity  of  its  tangent.  Thus,  CH  is  the  secant  of 
the  arc  AB,  or  of  its  supplement  BDE, 

6.  The  cosecant  of  an  arc,  is  the  secant  of  the  complement. 
Thus,  CL,  the  secant  of  BD,  is  the  cosecant  of  AB. 

6.  The  versed  sine  of  an  arc  is  the  difference  betwefm  the  cosine 
and  the  radius  ;  that  is,  AF  is  the  versed  sine  of  the  arc  AB,  and 
DK  is  the  versed  sine  of  the  arc  BD. 

For  the  sake  of  brevity  these  technical  terms  are  contracted  thus : 
for  sine  AB,  we  write  sin.AB,  for  cosine  AB,  we  write  cos.AB, 
for  tangent  AB,  we  write  tan.AB,  &c. 


D      LA 

a 

,1 

Fj 

'■/J 

.4- 

I 

/— ^  1 

•138  ELEMENTSOF 

From  the  preceding  definitions  we  deduce  the  following  obvious 
consequences : 

1st,  That  when  the  arc  AB,  becomes  so  small  as  to  call  it 
nothing,  its  sine  tangent  and  versed  sine  are  also  nothing,  and  its 
secant  and  cosine  are  each  equal  to  radius. 

2d,  The  sine  and  versed  sine  of  a  quadrant  are  each  equal  to  thf 
radius  ;  its  cosine  is  zero,  and  its  secant  and  tangent  are  infinite. 

3d,  The  chord  of  an  arc  is  twice  the  sine  of  half  the  arc.  Tims 
the  chord  BG,  is  double  of  the  sine  BF. 

4th,  The  sine  and  cosine  of  any  arc  form  the  two  sides  of  a 
right  angled  triangle,  which  has  a  radius  for  its  hypotenuse.  Thus, 
CF,  and  FBy  are  the  two  sides  of  the  right  angled  triangle  CFB, 

Also,  the  radius  and  the  tangent  always  form  the  two  sides  of  a 
right  angled  triangle  which  has  the  secant  of  the  arc  for  its  hypo- 
tenuse.     This  we  observe  from  the  right  angled  triangle  CAR. 

To  express  these  relations  analytically,  we  write 
sin.2+cos.2=i22  (1) 

i^'-ftan.'rizsec.'  (2) 

From  the  two  equiangular  triangles  CFB,  CAR,  we  have 
CF:FB=CA:AR 

That  is,         .        cos. :  sin.=i? :  tan.  tan.= '      (S) 

cos.       ^   ' 

Also,  .         CF:CB=CA:CR 

That  is,         .  cos  :  i?=i2 :  sec.         cos.  sec.=i2*       (4) 

The  two  equiangular  triangles  CAR,  CDL.  give 
CA'.AR=DL:DC 

That  is,         .  i? :  tan.=cot :  i2  tan.  cot.=i2^     (5) 

Also,  .         CF'.FB=DL:DC 

That  is,         .         COS. :  sin.=cot :  i2       cos.  i?=sin.  cot.     (6) 

By  observing  (4)  and  (5),  we  find  that 

cos.  sec.=tan.  cot.  (7) 

Or,  .         COS.  :tan.=cot.  :sec. 

The  ratios  between  the  various  trigonometrical  lines  are  always  the 
same  for  the  same  arc,  whatever  be  the  length  of  the  radius  ;  and 
therefore,  we  may  assume  radius  of  any  length  to  suit  our  conven- 
ience ;  and  the  preceding  equations  will  be  more  concise,  and  more 


PLANE    TRIGONOMETRY.  139- 

readily  applied,  by  making  radius  equal  unity.  This  supposition 
being  made,  the  preceding  becomes 

sin*^+cos.^=l  (1) 

l+tan.2=sec.^  (2) 

taii.==?l^     (3)  cos.=  —  (4) 

cos.     ^  '  sec.  ^  ' 

tan.= — -     (6)  cos. = sin.  cot.     (6) 

The  center  of  the  circle  is  considered  the  absolute  zero  point,  and 
the  different  directions  from  this  point  are  designated  by  the  different 
signs  +  and  — .  On  the  right  of  C7,  toward  A,  is  commonly 
marked  plus  (+),  then  the  other  direction,  toward  E,  is  necessarily 
minus  ( — ).     Above  AE\&  called  (+),  below  that  line  ( — ). 

If  we  conceive  an  arc  to  commence  at  Ay  and  increase  contin- 
uously around  the  whole  circle  in  the  direction  of  ABD,  then  the 
following  table  will  show  the  mutations  of  the  signs.  , 

sin.         COS.        tan.         cot         sec.      cosec.      vers. 

1st  quadrant.    -{-        +        +        4"        4*        -|-         "h 

2d"  -I-—.        —        —        —        +         -I- 

,3d"  ^        —        +        +        —        —        + 

4tJi       «  _        +        _        —        +        —        +/ 

PROPOSITION   1. 

The  chiyrd  of  60°  and  the  tangent  45°  are  each  equal  to  radius; 
the  sine  of  30°  the  versed  sine  of  60°  and  the  cosine  of  60°  are  each 
equal  to  half  the  radius. 

(The  first  truth  is  proved  in  problem  15,  book  1). 

On  C=,  as  radius,  describe  a  quadrant ;  take  -4i>=45°,  AB 
=60°,  and  ^^=90°,  then  i?j5'=30°. 

Join  ABy  CB,  and  draw  Bn,  perpendicular  to  CA,  Draw  Bm, 
parallel  to  AC.     Make  the  angle  C^^=90°,  and  draw  CBH. 

In  the  A  ABC,  the  angle  A  CB =60° 
by  hypothesis ;  therefore,  the  sum  of  the 
other  two  angles  is  (180— -60) =120°.  But 
CB=  CA,  hence  the  angle  CBA=  the  angle 
CAB,  (th.  1 5  b.  1 ) ,  and  as  the  sum  of  the  two 
is  120°,  each  one  must  be  60°;  therefore, 
each  of  the  angles  of  triangle  ABC,  is  60° 


140 


ELEMENTS    OF 


and  the  sides  opposite  to  equal  angles  are  equal  •  that  is,  AB,  the 
chord  of  60°,  is  equal  to  CA,  the  radius. 

In  the  A  CAH,  the  angle  CAH'is  a  right  angle ;  and  by  hypoth- 
esis, A  CH.  is  half  a  right  angle  ;  therefore,  AHC,  is  also  half  a  right 
angle ;  consequently,  AH=ACy  the  tangent  of  45°=  the  radius. 

By  th.  15,  book  1,  cor.  Cn=^nA;  that  is,  the  cosine  and  versed 
sine  of  60°  are  each  equal  to  the  half  of  the  radius.  As  Bn  and 
£!C  are  perpendicular  to  ^(7,  they  are  parallel,  and  Bm  is  made 
parallel  to  Cn;  therefore,  Bm==Cn,  or  the  sine  30°,  is  the  half  of 
radius. 

PROPOSITION     2. 

GHven  the  sine  and  cosine  of  two  arcs  to  find  the  sine  and  cosine  of 
the  sum,  and  difference  of  the  same  arcs  expressed  hy  the  sines  and  co- 
sines of  the  separate  arcs. 

Let  O  be  the  center  of  the  circle,  CD,  the 
greater  arc  which  we  shall  designate  by  a, 
and  DFy  a  less  arc,  that  we  designate  by  h. 

Then  by  the  definitions  of  sines  and  co- 
sines, 2>0=sin.a;    00=cos.a;   FI=sm.b; 
GI=:cos.b.     We  are  to  find  FM,  which  is 
=sm.(a-\-b);  GM=cos.(a-{-b); 
I!P=sm.(a--b);   OF=cos.(a—b). 

Because  /iV  is  parallel  to  DO,  the  two  As  GI) 0,  GIN",  are 
equiangular  and  similar.  Also,  the  A  FBI,  is  similar  to  GIN; 
for  the  angle  FIQ,  is  a  right  angle  ;  so  is  HIN;  and,  from  these 
two  equals  take  away  the  common  angle  HIL,  leaving  the  angle 
FIH=  GIN.  The  angles  at  H  and  N,  are  right  angles  ;  therefore, 
the  A  FHI,  is  equiangular,  and  similar  to  the  A  GIN,  and,  of 
course,  to  the  A  GD  0;  and  the  side  HI,  is  homologous  to  IN, 
and  D  0. 

Again,  as  FI=IE,  and  IK,  parallel  to  FM, 
FH^IK,  and  HI=KE. 

By  similar  triangles  we  have 

GD'.DO=GI:IN, 

T>     '  I    T\T  TUT    sin.a  cos  .J 

jB  :  sm.a=cos.6 :  iiv,   or  IN= 


That  is. 
Also, 


B 


GD:GO=FI:FB 


PLANE    TRIGONOMETRY. 


141 


That  is. 
Also, 
That  is. 
Also, 
That  is. 


B:co3.a=sia.b:FB',  or  Fir= 

OD'.ao^Gi'.GJsr 


R :  cos.a=cos.6 :  (?i\^,  or  (?ir= 
OD:B0=FI'.IH 
It :  sin.a=sin.5 :  IE,   or  IH= 


cos.a  sin.6 
R 

cos.a  cos. J 


M 


sin.a  sin.J 


It 


Bj  adding  the  first  and  second  of  these  equations,  we  have 

m-{'FJI=FM=sm.(a+h) 

r^,^    ,  .                   •      /    ,  ,x      sin.tt  cos.J+cos.a  sin.5 
That  IS,        .      sm.  (a+b)— ~ — 

By  subtracting  the  second  from  the  first,  we  have 

,       _.     sin.a  C0S.5— cos.a  sin.J 
sm.  (a— 5)= ^ 

By  subtracting  the  fourth  from  the  third,  we  have 

02^— IR=OM=  COS. (a-^h)  for  the  first  member, 
cos.a  cos.5 — sin.a  sin.b 


Hence, 


cos.(a+^)= 


It 


By  adding  the  third  and  fourth,  we  have 

GK+IH=  QX-{-NP=  GP=:cos.{a—b) 


Hence, 


COS.  (a — b) 


_cos.a  COS.  5-4-siii-<st  sin. J 


(^) 


Collecting  these  four  expressions,  and  considering  the  radius 
unity,  we  have 

sin.(a+5)=sin.a  cos.J+cos.a  sin.5  (7) 
sm.(a---b):=sm.a  cos.5— cos.a  sin.5  f  8) 
cos.(a+5)=cos.acos.i — sin.a  sin.6  (9) 
cos.(a— j)=cos.a  cos.^+sin.a  sin.5       (10) 

Formula  (A),  accomplishes  the  objects  of  the  proposition,  and 
from  these  equations  many  useful  and  important  deductions  can  be 
made.     The  following,  are  the  most  essential : 

By  adding  (7)  to  (8),  we  have  (11);  subtracting  (8)  from  (7), 
gives  (12).  Also,  (9)+(10)  gives  (13);  (9)  taken  from  (10) 
gives  (14). 

sin.(a-|-5)+sin.(a — 5)=2sin.a  cos  5 
smJa-\-bS — sin. (a — 5)=2cos.a  sin.  5 
cos.(a4-^)+cos.(a — 5)=2cos.o  C08.b 
cos.(a— i) — cQS.(a+6)=2sin.  a  sin.6 


(£) 


142 


ELEMENTS    OF 


If  we  put  a-]r-h=A,  and  a — h=B,  then  (11)  becomes  (15), 
(12)  becomes  (16),  13  becomes  (17),  and  (14)  becomes  (18). 


(0) 


sm.-4-|-sm.^=2sm.  f  — - —  I  cos.  (  — - —  1 

'     A       '     -n     ^        (^-^B\    .       /A—B\ 
sm,A — ^sm.-D=:2cos.  [  — - —  J  sm.   f  — ---  J 

At         DO        f  ^+^  \  /  ^— ^  \ 

cos.-4+cos.^=2cos.  f  — - —  J  cos.  i  — —  -  j 

cos.^ — cos.-4=2sin.  [  — —  -  j  sin.  (  — — -    j 


(15) 
(16) 

(17) 
(18) 


sm. 


If  we  divide  (15)  by  (16),  (observing  that  =tan.   and 

COS. 

cos.  1 

-T— ^=cot.=:- —  as  we  learn  by  equations  (6)  and  (5)  trigonome- 


try), we  shall  have 


sin.-4+sin.i5 


s,n.(— -j      cos.(— ^)     tan.(— -) 


sin.^ — sin.^ 


(19) 


COS. 


Whence, 

,      fA-hB\   ^      iA—B\ 

sm.^+sin.  J?  :  sm.-4 — sm.^=tan.  ^  — g—  I '-  tan.  I  — - —  I 

or  in  words.  The  sum  of  the  sines  of  any  two  arcs  is  to  the  differ- 
ence of  the  same  sines y  as  the  tangent  of  the  half  sum  of  the  same  arcs 
is  to  the  tangent  of  half  their  difference. 

By  operating  in  the  same  way  with  the  diflferent  equations  in  for- 
mula (  C),  we  find, 

f   sin.^+sin.^  __^ 
cos.u4-|-cos.j5 

sin.^+sin.^         ^   /  A — B  \ 
^=cot.  (  — -  ) 


{!>) 


:-..(^^) 


COS.^ COS 

sin.-4 — sin.^ 


cos.^-f-cos.-S  \      2      / 

sin.^ — sin.5 /  A-^B  \ 

cos.-B— cos.^  \      2      / 


cos.-4+cos.-B 
cos.jB — C0S.-4' 


tan 


A—B 


r-^i 


(20) 
(21) 
(22) 
(23) 

(24) 


PLANE    TRIGONOMETRY  143 

These  equations  are  all  true,  whatever  be  the  value  of  the  arcs 
designated  by  A  and  B;  we  may  therefore,  assign  any  possible 
value  to  either  of  them,  and  if  in  equations  (20),  (21)  and  (24), 
we  make  B=^  0,  we  shall  have, 

sin.^        ,-4         1  ,^^. 

=tan.-= r,  (25) 


1+C0S..4  2      cot.^-4 

sin.-4  A         1  .^. 

.j=coV=— — .  (26) 


1— C0S.-4  2      tan.^-4 

l-i-cos.-4     cot.-J^  1 

1 — cos.J^     tan.^A     tan.^^. 
If  we  now  turn  back  to  formula  (-4),  and  divide  equation  (7)  by 

),  and  (8)  b; 

we  shall  have, 


l4-cos.^^cot4^_ 1^  _ 

1— cos.^     tan4-4     tan.^^^  ^     ^ 


sin 
(9),  and  (8)  by  (10),  observing  at  the  same  time,  that  — ^=tan, 

cos* 


,    ,  ,.     sin  a  cos.J+cos.a  sin.6 

tan.(a+ J) = t — ^  —, 

^        ^     cos.a  cos.o — sm.a  sm.o 

,       ,.     sin.a  cos.i — cos.a  sin.5 

tan.(a — 6)= rr-- HE 

^         ^     cos.a  cos.o-f-sm.a  sm.o 

By  dividing  the  numerators   and  denominators  of  the  second 

members  of  these  equations  by  (cos.a  cos. 5),  we  find, 

sin.a  cos.5     cos.a  sin.5 

,    I  T\     cos.a  cos.6     cos.a  C0S.6      tan.a+tan.J 

tan.(a+5)= j-—. ^~.=z — r ; — i.     (28) 

^         ^     cos.a  cos.o     sm.a  sm.6     1 — tan.a  tan.o      ^     ' 

cos.a  cos,b     cos.a  cos.6 
sin.a  C0S.5     cos.a  sin.5 

,       ,.     cos.a  C0S.5     cos.a  cos .5     tan.a — tan.J        ,  ^. 

tan.(a— ^)= r — r— v-  7=r-r. ; — r     (29) 

^       -       cos.a  cos.o     sm.a  sm.o     1 -[-tan.a  tan.o      ^     ^ 

cos.a  COS. 6     cos.a  cos. 6 

If  in  equation  (11),  formula  (jB),  we  make  a=b,  we  shall  have, 

sin.2a=2sin.a  cos.a         (30) 
Making  the  same  hypothesis  in  equation  (13),  gives, 

cos.2a4-l=2cos^a  (31) 

The  same  hypothesis  reduces  equation  (14),  to 
1 — cos.2a=2sin^a  (32) 

The  same  hypothesis  reduces  equation  (28),  to 

2tan.a  ,„„. 


144  ELEMENTS    OF 

If  we  substitute  a  for  2a  in  (31)  and  (32),  we  shall  have 
l-|-cos.a=2  cos.'^a.  (34) 

and  1 — cos.a— 2  sin.^^a.  (35) 

Recurring  again  to  formula  {B),  we  have,  by  transposing 
sin.(a+i)=2sin.a  cos.i — sin.(a — b) 
sin.(a-|-i)=2cos.a  sin.54-sin.(a — b) 
If,  in  the  first  of  these  expressions,  we  make  a =30°,  2sin.a  will 
equal  radius,  or  unity;  and  if  in  the  second  we  make  «=60°,  2cos.a 
will  also  equal  unity;  these  expressions  then  become, 

sin.(30°-|-5)=cos.5— sin.(30°— 6)  (36) 

And     .     .     sin.(60°+i)=sin.5-fsin.(60— °i)  (37) 

The  sines  may  be  easily  continued  to  60°,  by  equation  (36), 
when  the  sines  and  cosines  of  all  arcs  below  30°  have  been  com- 
puted; then,  by  equation  (37),  the  sines  can  be  readily  run  up  to  90°. 
The  foregoing  equations  might  have  been  obtained  geometrically, 
but  not  so  easily  and  concisely. 

ON    THE    CONSTRUCTION    OF   TABLES    OF 
SINES,   TANGENTS,    &c. 

To  explain  this,  we  refer  at  once  to  Table  II,  which  contains  loga- 
rithmic sines,  and  tangents,  and  also  natural  sines  and  cosines.  The 
natural  sines  are  made  to  the  radius  of  unity;  and,  of  course,  any  par- 
ticular sine  is  a  decimal  fraction,  expressed  by  natural  numbers.  The 
logarithm  of  any  natural  sine,  with  its  index  increased  by  1 0,  will  give 
the  logarithmic  sine.  Thus,  the  natural  sine  of  3°  is  .052336 
The  logarithm  of  this  decimal  is        .         .         .  —2.718800 

To  which  add 10. 

The  logarithmic  sine  of  3°  is,  therefore,     .         .  8.718800 

In  this  manner  we  may  find  the  logarithmic  sine  of  any  other  arc, 
when  we  have  the  natural  sine  of  the  same  arc. 

If  the  natural  sines  and  logarithmic  sines  were  on  the  same  radius, 
the  logarithm  of  the  natural  sine  would  be  the  logarithmic  sine,  at  once, 
without  any  increase  of  the  index. 

The  radius  for  the  logarithmic  sines,  is  arbitrarily  taken  so  large 
that  the  index  of  its  logarithm  is  10.  It  might  have  been  more  or  less  ; 
but,  by  common  consent,  it  is  settled  at  this  value  ;  so  that  the  sines 
of  the  smallest  arcs  ever  used  shall  not  have  a  negative  index. 

In  our  preceding  equations,  sin.a,  cos.a,  &c.,  referred  to  natural 
sines;  and  by  such  equations  we  determine  their  values  in  natural  num- 
bers ;  and  these  numbers  are  put  in  the  table,  as  seen  in  table  2,  under 
the  heads  of  not.  sine,  and  nat.  cosine. 


PLANE    TRIGONOMETRY.  145 

To  commence  computation,  we  must  know  the  sine  or  cosine  of  some 
known  arc  ;  and  we  do  know  the  sine  and  cosine  of  30°.  The  sine  of 
30°  is  ^  (prop.  1,  trig.),  and,  hence,  cos'.  30°=1 — i  (eq.  (1)  trig.); 
or,  COS.  30°=^<,y3.     Now  put  2a=30°,  and  equation  (30)  gives 

2sin.l5o  cos.l6°=0.6.  (n) 

Eq.  (1)  gives    .        .    cos.2i6°-f  sin.2i5=l.  (n) 

By  adding  (w)  to  (w),  and  extracting  square  root,  we  obtain, 
cos.l6°-f-  sin.l5o=Vl-^=l-22474487.         (p) 

By  subtracting  (m)  from  <7i),  and  extracting  square  root, 

C0S.160—  sin.l5°==V0.6=0.707l0678         (q) 

Sub.  (q)  from  (p)  gives        2sin.l5°=0.5 1763709. 

Again,  put  2a=15°,  and  in  like  manner  apply  equations  (30)  and 
(1),  and  we  can  have  the  sine  and  cosine  of  7°  30',  and  thus  we  may 
bisect  as  many  times  as  we  please,  but  when  we  get  down  to  any  arc 
under  1',  we  can  compute  the  sines  by  direct  proportion. 

Also,  by  theorems  3  and  4,  book  6,  the  semicircumference  of  a  circle 
whose  radius  is  unity,  is  3.14169265;  this,  divided  by  10800,  the  num- 
ber of  minutes  in  180°,  will  give  .0002908882  for  the  length  of  the  sine 
or  arc  of  one  minute.  The  logarithm  of  this  number,  with  its  index 
increased  by  10,  gives  6.463726,  the  log.  sign  of  1',  which  is  found  in 
the  table. 

Having  the  sine  and  cosine  of  I',  we  can  find  the  sine  and  cosine  of 
2' by  equation  (30); 

That  is,        .        .       sin.2a=2  sin.a  cos.a 

Or,        .        .        .       sin.2'=2  sin.l'cos.l' 

For  the  sine  of  3',  and  every  succeeding  minute,  we  apply  equation 
(11),  making  a=2',  and  b=l'; 

That  is,        .        .        sin.3'=2  sin.2' cos.l — sin.l' 

Having  the  sine  of  3',  we  obtain  the  sine  of  4'  by  the  application  of 
the  same  equation  ;  that  is,  by  making  a=Z\  and  b=l; 

Then,  .        .        .        sin.4'=2  sin. 3'  cos.l —  sin.2' 

sin.5'=2  sin. 4'  cos.l —  sin. 3'     &c.,  &c. 

When  the  sine  of  any  arc  is  known,  its  cosine  is  readily  determined 
by  the  following  formula,  which  is,  in  substance,  equation  (1), 
trigonometry.         .         .      cos.=^(l-{-sin.)(l — sin.) 

When  the  sine  and  cosine  of  any  arc  are  known,  the  sine  and  cosine 
of  its  double,  are  found  from  equation  (30);  and  thus,  from  equations 
(30),  (11),  and  (1),  the  sines  and  cosines  of  all  arcs  can  be  determined. 

When  the  sine  and  cosine  of  an  archavebeen  determined  through  a 
series  of  operations,  the  accuracy  of  the  results  should  be  tested  by 
10 


146  ELEMENTS    or 

equation  (12)  or  (14),  or  by  some  other  equation  independent  of  former 
operations;  and  if  the  two  results  agree,  they  may  be  regarded  as 
accurate. 

One  independent  method  will  be  found  by  applying  theorem  5,  book 
6.  In  that  theorem  we  find  the  chord  of  20°  is  .347296  ;  the  natural 
sine,  then,  of  10°,  is  .173648.  Taken,  the  chord  of  20°,  and  trisecting 
the  arc  by  the  same  problem,  we  find  the  chord  of  6°  40'  to  be  .11628; 
and,  of  course,  the  natural  sine  of  3°  20'  is  .05814;  and  thus,  by 
successive  trisections  we  can  obtain  the  sines,  and  of  course  the  cosines 
of  certain  arcs  ;  and  when  we  arrive  at  very  small  arcs,  we  can  com- 
pute their  increase  or  decrease  by  direct  proportion.* 

Now,  if  the  sine  of  an  arc  computed  through  successive  trisections, 
agrees  with  the  sine  of  the  same  arc  computed  through  successive 
bisections,  we  must,  of  course,  regard  the  result  as  accurate. 

When  we  have  the  sines  and  cosines  of  an  arc,  the  tangent  and  co- 
tangent are  found   by  (3)    tan.= — ^^*     (6)    cot.= ^-;    and  the 

cos.  sin. 

jfJ2 

secant  is  found  by  equation  (4) ;  that  is,   sec.= 

COS. 

For  example,  the  logarithmic  sine  of  6°,  is  9.019235,  and  its  cosine 
9.997614.  From  these  it  is  required  to  find  the  tangent,  cotangent, 
and  secant. 

JRsin.        .        .         19.019236 

Cos.  .  subtract    9.997614 


Tan.  is 
jRcos. 
Sin.  . 

Cotan.  is 
JRMs 

Cos.  . 

Secant  is 


9.021621 
19.997614 

subtract  9.01923^ 
10.978379 
20.000000 

subtract  9.997674 
10.002326 


•  Thus,  from  theorem  4,  book  5,  we  find  the  chord  of  28'  7"  30"'  to  be 
.008181208  ;  and  wishing  to  take  away  7"  30'",  we  do  it  by  proportion,  as 
follows.     The  sine  of  1'  or  60"  is  .0002908882. 

Therefore,        .  60  :  7J=.0002908882 


Or,  .        .        .8:1  =.0002908882  : 

.000036461 

The  chord  of  28' 7"  30'"  is 

. 

.008181208 

of        7"  30'"  is 

. 

.000036461 

of  SS'              is 

,        , 

.008144747 

The  natural  sine  of  14'    is 

. 

.004072373 

Now  we  may  halve  or  doubU  thli  8in«  by  equation  (30). 


PLANE    TRIGONOMETRY  147 

The  secants  and  cosecants  of  arcs  are  not  given  in  our  table,  because 
they  are  very  little  used  in  practice  ;  and  if  any  particular  secant  is 
required,  it  can  be  determined  by  subtracting  the  cosine  from  20  ;  and 
the  cosecant  can  be  found  by  subtracting  the  sine  from  20. 


PROPOSITION    3. 

In  any  right  angled  plane  triangle^  we  may  have  the  follomng 
proportions  : 

1st.  As  the  hypoteniLse  is  to  either  side,  so  is  the  radius  to  the  sine 
of  the  angle  opposite  to  that  side. 

2d.  As  one  side  is  to  the  other  side,  so  is  the  radius  to  the  tangent 
of  the  angle  adjacent  to  the  first-mentioned  side. 

3d.  As  one  side  is  to  the  hypotenuse,  so  is  radium  to  the  secant  of 
the  angle  adjacent  to  that  side. 

Let  CAB  represent  any  right 
angled  triangle,  right  angled  at  A. 
AB  and  AC  are  called  the  sides 
of  the  A,  and  CB  is  called  the 
hypotenuse. 

(Here,  and  in  all  cases  hereafter,  we  shall  represent  the  angles  of  a  triangle 
by  the  large  letters  A,  B,  C,  and  the  sides  opposite  to  them,  by  the  small  letters 
a,  h,  c.) 

From  either  acute  angle,  as  C,  take  any  distance,  as  CD,  greater 
or  less  than  CB,  and  describe  the  arc  DE.  This  arc  measures 
the  angle  C.  From  D,  draw  DF  parallel  to  BA;  and  from  E, 
draw  EO,  also  parallel  to  BA  or  DF. 

By  the  definitions  of  sines,  tangents,  and  secants,  DF  is  the  sine 
of  the  angle  C;  EQ-  is  the  tangent,  CQ  the  secant,  and  CF  the 
cosine. 

Now,  by  proportional  triangles  we  have, 

CB  :  BA=CD  :  DF    or,  a  :  c=B  :  sin. (7) 

CA  :.AB=CE  :  EG    or,  b  :  c=E  :  tan. (7  V  Q.  E.  D. 

CA  :  CB=^CE  :  CG    or,  h  :  az=zR  :  sec.Cj 

Scholium.  If  the  hypotenuse  of  a  triangle  is  made  radius,  one 
side  is  the  sine  of  the  angle  opposite  to  it,  and  the  other  side  is  the 
cosine  of  the  same  angle.     This  is  obvious  from  the  triangle  CDF. 


148  ELEMENTS    OF 

PROPOSITION    4. 

In  any  triangle f  the  sines  of  the  angles  are  to  erne  another  as  the 
sides  opposite  to  them. 

Let  ABC  be  any  tri- 
angle. From  the  points 
A  and  B,  as  centers,  with 
any  radius,  describe  the 
arcs  measuring  these  an- 
gles, and  drawjt?a,  CD, 
and  m»,  perpendicular  to  AB, 

Then,        .        .  pa=^m.A,  mn=Bm.B 

By  the  similar  As,  Apa  and  A  CD,  we  have, 

B  :  sin.^=5  :  CD;  or,  R{CD)=h  Bm,A    (1) 

By  the  sinular  As  Bmn  and  BCD,  we  have, 

R  :  sin.^=a  :  CD;  or,  R{CD)=asm.B    (2) 
By  equating  the  second  members  of  equations  (1)  and  (2). 

h  sin.-4=a  sin.^. 
.  Hence,      ,  sin.-4  :  sin.-B=a  :h  \  O  E  D 

Or,  .        .  o  :  5=sin  A  :  sin.  B) 

Scholium  1.  When  either  angle  is  90°,  its  sine  is  radius. 

Scholium  2.  When  CB  is  less  than  A  C,  and  the  angle  B^  acute, 
the  triangle  is  represented  by  A  CB,  When  the  angle  B  becomes 
B' ,  it  is  obtuse,  and  the  triangle  is  A  CB' ;  but  the  proportion  is 
equally  true  with  either  triangle ;  for  the  angle  CB'D=  CBA, 
and  the  sine  of  CB'D  is  the  same  as  the  sine  of  AB'C.  In  prac- 
tice we  can  determine  which  of  these  triangles  is  proposed  by  the 
side  AB,  being  greater  or  less  than  A  C;  or,  by  the  angle  at  the 
vertex  C,  being  large  as  A  CB,  or  small  as  A  CB', 

In  the  solitary  case  in  which  A  C,  CB,  and  the  angle  A,  are  given, 
and  CB  less  than  A  C,  we  can  determine  both  of  the  As  A  CB 
and  A  CB';  and  then  we  surely  have  the  right  one. 

PROPOSITION    5. 

J^  from  any  angle  of  a  triangle,  a  perpendicular  he  let  fall  on  the 
opposite  side,  or  hose,  the  tangents  of  the  segments  of  the  angle  are  to 
one  another  as  the  segments  of  the  base. 


PLANE    TRIGONOMETRY.  149 

Let  ABC  be  the  triangle.  Let  fall  the 
perpendicular  CD,  on  the  side  AB, 

Take  any  radius,  as  Criy  and  describe 
the  arc  which  measures  the  angle  C. 
From  w,  draw  qnp  parallel  to  AB.  Then 
it  is  obvious  that  np  is  the  tangent  of  the 
angle  D  CBy  and  nq  is  the  tangent  of  the  angle  A  CD. 

Now,  by  reason  of  the  parallels  AB  and  qp,  we  have, 
qn  :  np=^AD  :  DB 

That  is,     \An.ACD  :  ian.DCBz=AD  :  DB         Q.  E,  D. 

PROPOSITION     6. 

If  a  perpendicular  he  let  fall  from  any  angle  of  a  triangle  to  its  op- 
*^site  side  or  base,  this  base  is  to  the  sum  of  the  other  two  sides,  as  the 
difference  of  the  sides  is  to  the  difference  of  the  segments  of  the  base. 
(See  figure  to  proposition  5.) 

Let  AB  be  the  base,  and  from  (7,  as  a  center,  with  the  shorter 
side  as  radius,  describe  the  circle,  cutting  AB  in  0,AC  in  F,  and 
produce  AG  to  £J. 

It  is  obvious  that  AB  is  the  sum  of  the  sides  A  C  and  CB,  and 
AF  is  their  difference. 

Also,  AD  is  one  segment  of  the  base  made  by  the  perpendicular, 
and  BD=DG  is  the  other;  therefore,  the  difference  of  the  seg- 
ments is  A  G. 

As  .^  is  a  point  without  a  circle,  by  theorem  18,  book  3,  we  have, 
AFXAF=ABXAG 

Hence,         .        .     AB  :  AF=AF :  AG        Q.  K  D. 

PROPOSITION     7. 

The  sum  of  any  two  sides  of  a  triangle,  is  to  their  difference,  as 
the  tangent  of  the  half  sum  of  the  angles  opposite  to  these  sides,  to 
the  tangent  of  half  their  difference. 

Let  ABC  be  any  plane   triangle.     Then, 
by  proposition  4,  trigonometry,  we  have, 

CB  :^(7=sin.  A  :  sin.B 
Hence, 
CB-^-AC:  CB—AC=sm.A+sm.B  :  sin.-4--sin.^  (th.  9  b.  2) 


150  ELEMENTS    OF 

But,  tan.  I  — ~ —  J  :  tan.  I  — - —  J  =sin.-4+sm.j5  :  sin.^ — sin.B 
(eq.  (19),  trig.) 

Comparing  the  two  latter  proportions  (th.  6,  b.  2),  we  have, 

Ci5+^(7  :  (7.5-^a=tan.(:^^):tan.(^-]  Q.  K  D, 


PROPOSITION    8. 

Given  the  three  sides  of  any  plane  triangle,  to  find  some  relation 
which  tliey  rmist  hear  to  the  sines  and  cosines  of  the  respective  angles. 

Let-4J5C7bethe 
triangle,  and  let  the 
perpendicular  fall 
either  upon,  or 
without  the  base, 
as  shown  in  the 
figures ;  and  by 
recurring  to  theorem  38,  book  1,  we  shall  find 


C    a    jFJ      X     JJ 


CD: 


2a 


(0 


Kow,  by  proposition  3,  trigonometry,  we  have, 
JB  :  COS.  (7=6  :  CD 
h  COS.  C 


Therefore, 


CD: 


R 


(2) 


^    R{a''-\-b'—c^) 

COS.  C7=— i — -— ^ 

2ab 


Equating  these  two  values  of  CDy  and  reducing,  we  have. 

In  this  expression  we  observe  that  the  part  of  the  numerator 
which  has  the  minus  sign,  is  the  side  opposite  to  the  angle ;  and 
that  the  denominator  is  twice  the  rectangle  of  the  sides  adjacent 
to  the  angle.  From  these  observations  we  at  once  draw  the  fol- 
lowing expressions  for  the  cosine  -4,  and  cosine  B. 


Thus, 


C0S.-4 


cos 


2bc 
2ac 


(n) 
(P) 


PLANE    TRIGONOMETRY.  151 

As  these  expressions  are  not  convenient  for  logarithmic  compu- 
tation, we  modify  them  as  follows  : 

If  we  put  2a=A,  in  equation  (31),  we  hare, 
C0S.-4+ 1=2  cos.^  ^A 

In  the  preceding  expression  (»),  if  we  consider  radius,  imitjr, 
and  add  1  to  both  members,  we  shall  have. 


cos.^+l=l+ 


2bc 


Therefore,         2  cos.'  |^r=?^f±^t^-Z:i 

2bc 
Considering  (b+c  )  as  one  quantity,  and  observing  that  we  have 
the  diflference  of  two  squares,  therefore 
(J+c)2— a2=(6+c+a)(5+c-- a);  but  (6+c— a)=5-f  c+a— 2a 

Hence,        .     2  cos.' i^=(^+^lg±f±^=5L) 

Or,     ,        .         cos/ ^-4= — — 

By  putting  — - — =s,  and  extracting  square  root,  the  final 
result  for  radius  unity,  is 


1  A       Ks—a) 
eos.i^=^-A_^-^ 

For  any  other  radius  we  must  write, 

,   .       IRh(s—^ 
cos.J^=^-^ 

«    .  i.                           ,  T>       lRh(s—h) 
By  mference,  cos4^=  -yj ^ ^ 

Also,        .        .     COS. iC=yJ -T-—^ 

In  every  triangle,  the  sum  of  the  three  angles  must  equal  180°;  and 
if  one  of  the  angles  is  small,  the  other  two  must  be  comparatively 
large;  if  two  of  them  are  small,  the  third  one  must  be  large.  The 
greater  angle  is  always  opposite  the  greater  side  ;  hence,  by  merely 
inspecting  the  given  sides,  any  person  can  decide  at  once  which  is  the 
greater  angle ;  and  of  the  three  preceding  equations,  that  one  should 
be  taken  which  applies  to  the  greater  angle,  whether  that  be  the  par- 
ticular angle  required  or  not;  because  the  equations  bring  out  the 


152  ELEMENTS    OF 

cosines  to  the  angles ;  and  the  cosines,  to  very  small  arcs  vary  so  slowly, 
that  it  may  be  impossible  to  decide,  with  sufficient  numerical  accuracy 
to  what  particular  arc  the  cosine  belongs.  For  instance,  the  cosine 
9.999999,  carried  to  the  table,  applies  to  several  arcs  ;  and,  of  course, 
we  should  not  know  which  one  to  take  ;  but  this  difficulty  does  not  exist 
when  the  angle  is  large  ;  therefore,  compute  the  largest  angle  first, 
and  then  compute  the  other  angles  by  proposition  4. 

But  we  can  deduce  an  expression  for  the  sine  of  any  of  the  angles, 
as  well  as  the  cosine.     It  is  done  as  follows  : 

EQUATIONS    FOR    THE   SINES   OF    THE    ANGLES. 
Resuming  equation  (m),  and  considering  radius,  unity,  we  have, 

COS.  C= — -—z 

2aJb 

Subtracting  each  member  of  this  equation  from  1,  gives 

'— ^=-(^-)     (0 

Making  2a=(7,  in  equation  (32),  then  a=^0. 

And         .         .   1— cos.(7=2  sin.2^(7  (2) 

Equating  the  right  hand  members  of  (1)  and  (2), 

2  sin.^^(7= 


2al>—a*-^^-\-c^ 


2ab 

2ab 

(c-\-h — a)(c'{-a — b) 

2ab 
/  c-\-b — a  \   /  c-{-a — 'b  \ 

.    ,,  ^      ^        2        I  I        2        / 
Or,     .        .         .  sm.>i(7= ^ 

But,   .        =— a   and =_X_J j 

2  2  2  2 

Put   .         — - — =8,    as  before  ;  then. 


By  taking  equation  (p),  and  operating  in  the  same  manner,  we 

have      .        .        .    sm.i£=Jl^H^ 
^        ^  ac 

From  (n)  .        .  6m.iA=J^'~^^['~^^ 

N  CO 


sm.iC=yJ'- 


PLANE    TRIGONOMETRY.  153 

The  preceding  results  are  for  radius  unity;  for  any  other 
radius,  we  must  multiply  by  the  number  of  units  in  such  radius. 
For  the  radius  of  the  tables,  we  write  B;  and  if  we  put  it  under 
the  radical  sign,  we  must  write  H^;  hence,  for  the  sines  corres- 
ponding with  our  logarithmic  table,  we  must  write  the  equations 

thus,      .        .        .  sin4^=^M^35^ 

3in4i?=^/5S3^ 

r_    IR\s-a){s-b) 
ah 

A  large  angle  should  not  be  determined  by  these  equations,  for 
the  same  reason  that  a  small  angle  should  not  be  determined  from 
an  equation  expressing  the  cosine. 

In  practice,  the  equations  for  cosme  are  more  generally  used, 
because  more  easily  applied. 

In  the  preceding  pages  we  have  gone  over  the  whole  ground  of 
theoretical  plane  trigonometry,  although  several  particulars  might 
have  been  enlarged  upon,  and  more  equations  in  relation  to  the 
combinations  of  the  trigonometrical  lines,  might  have  been  given  ; 
but  enough  has  been  given  to  solve  every  possible  case  that  can  arise 
in  the  practical  application  of  the  science  ;  but  to  show  more  clearly 
the  beauty  and  spirit  of  this  science,  and  to  redeem  a  promise,  we 
give  the  following  geometrical  dem(mstraiions  of  the  truths  expressed 
in  some  of  the  preceding  equations. 

From  C  as  the  center,  with  CA  as  the  radius,  describe  a  circle. 
Take  any  arc,  AB^  and  call  it  A;  AD  a  less  arc,  and  call  it  B;  then 
BD  is  the  difference  of  the  two  arcs,  and  must  be  designated  by 
(A—'B);AG=AB;  therefore,  DG=A-\-B;   UG=sm,A; 
(See  fig.  p.  154.)       Un=sm.B;  Gn=sin.A-\-sin.B; 
Bn=sm.A — sin.^. 
Fm=mD=^CHz=cos.B;  mn=cos.A; 

Therefore,     Fm-{-mn=cos.A-\-cos.B=Fn; 
mD — mn=cos.B — cos.^=wi>/ 

Because        .        NF^AD;  AB'\'NF^A-\-B; 
Therefore,     .        .  180°--(^-i-J5)=arc  i^^; 


154  ELEMENTSOF 

Or,       .         .         .         90°— (^-)=JarcJ?'5; 

But  the  chord  FB,  is  twice  the  sine  of  -J  arc  FB. 

That  is,         FB=^2sm,  (  90°— ^^^^  ]  =2cos.  (  ^^  \ 

The     angle     nGD=BFD,    because 
both  are  measured  by  one  half  of  the 

arc  BD;  that  is,  by    [  — —  -  ]  and  the 

two  triangles  OnD,  and  FnB  are  similar. 
The  angle  GFuy  is  measured  by 

In  the  triangle  FBOy  Fn  is  drawn  from  an  angle  perpendicular 
to  the  opposite  side ;  therefore,  by  Proposition  5,  we  have, 

Gn :  n^=tan.  GFn :  i2LW.BFn 

That  is,  sin.-4+sin.-B :  sin.-4 — sin.5=tan.  (         ■     ]  :  tan.  (  — — -  ] 

This  is  equation  (19). 

In  the  triangle  GnDy  we  have 

sm.90*»:i)6^sin.ni>(7:  Gn;  sm.nI)G=co8.nGD 

That  is,  1 :  2sin.  (  — ^  ]  =cos.  (  — "^  ]  ;  sin.^+sin.^ 

Or,         .        sin.-44-sin.5=2sin.  (  —- —  )  cos.  I  -~—  ) 
same  as  equation  (16). 

In  the  triangle  FnB,  we  have, 

sin.90  :  FB=sm.BFn :  Bn 

That  is,        1  :  2cos.  f  — -—  j  =sin.  [  -~  )  :  sin.^—sin.  J5 

Or,         .         .     sin.-4 — sm.^=2cos.  [  — ^-  )  sin.  (  -~-  ) 
same  as  equation  (16). 

In  the  triangle  FBn,  we  have, 

sin.90  :  FB=cos.BFn  :  Fn 

That  is,        1 :  2cos.  /  — g—  )  =cos.  (  -^-  ):co&.A-\'Coa,B 


PLANE    TRIGONOMETRY.  155 

Or,  cos.^4-cos.i?=2cos.  (     T"     )  cos.  [  — - —  J  same  as  equa- 
tion (17). 

In  the  triangle  OnD^  we  have, 

sin.90°  :  GD=sm.nGD  :  nD 

That  is,       .    1 :  2sin.  [  — - —  j  =sin.  j  — - —  J  :  cos.^ — cos.^, 

same  as  equation  (18). 

In  the  triangle  FGn,  we  have, 

sin.  OFn :  6^= cos.  OFn  :  Fn 

A-^-B  A-\-B 

That  is,     sin. — -—  :  sin.^+sin.jB=cos. — - — :cos.^4-cos.5 

Or,   (sin.-44-sin.^)cos.  I  — - —  J  =(cos.^+cos.^)sin.  [  — — --  j 

A+B 
•     ^  I    •     D     sm 
Or. 


sm  - 
sin.^H-sin.5_  ^     _.       ( A-\-B  \ 

^.^-fcos.j5~       3T^~        I  ~2~"  j 

COS. — - — 

2 


same  as  equation  (20). 

We  give  a  few  more  geometrical  demonstrations  from  the  follow- 
ing figure : 

Let  the  arc      AD=A;    then  i>6^=sin.^;     CO=cos.A; 
I>I=sm.^A;     ^i>=2sin.^^;  CI=cos.^A; 
CI=D  0;         DB=2D  (9=2cos.i^. 

The  angle  DBA,  is  measured  by  half 
AD;  that  is,  by  ^A. 

Also,         .  ADG==J)BA==iA. 

Now  in  the  triangle  BDO,  we  have, 
sm.I)BG:I)O=sm.90°  :BD 

That  is,  sin.-J^^  :  sin.^=l  :  2cos.-J-^ 

Or,  .  sin.^=2sin.i[--4cos.^^ 

same  as  equation  (30). 

In  the  same  triangle 

sin.90° :  BD=sm.BDG:BG;  sm.BDG=cos.DBG; 
That  is,   .      1  :  2cos.^^=cos.^^  :  l+cos.^ 
Or,  .         2cos^^-4=sl-fco8.-4,  same  as  equation  (34). 


156  KLEMENTSOF 

In  the  triangle  D  QA,  we  have, 

Bin.90°  :^i>=sin.6?i>^  :  OA 
That  is,    .      1 :  2sin4-4=6in4^  :  1 — cos.^ 
Or,  .         2sin.^^^=l — cos.^,  same  as  equation  (35). 

By  similar  triangles,  we  have, 

BA\AD=zAD\AQ 
That  is,    .      2 :  2sin.^^=2sin.^-4 :  versed  sin.-4 
Or,  •  versed  sin.-4=2sin.^^^. 


APPLICATION    OF    THE    PRINCIPLES    OF 
TRIGONOMETRY. 

Every  triangle  consists  of  six  parts;  three  sides,  and  three  angles  ; 
and  to  determine  all  the  parts,  three  of  them  must  be  given,  and  at 
least  (ym  of  these  parts  mtist  be  a  side,  because  two  triangles  may  have 
equal  angles,  and  their  sides  be  very  different  in  respect  to  magnitude 

In  right  angled  plane  triangles,  the  right  angle  is  always  given  ;  and 
if  two  other  parts,  and  one  a  side,  be  given,  it  will  be  sufficient  for  the 
complete  determination  of  all  the  other  parts. 

Before  the  invention  of  logarithms,  the  numerical  computations  for 
the  parts  of  a  triangle  were  all  made  by  arithmetical  proportion,  as  in 
the  rule  of  three,  through  the  help  of  natural  sines  and  cosines  ;  but 
the  operations,  in  many  cases,  were  extremely  laborious.  For  mere 
curiosity,  we  will  use  natural  sines  to  solve  the  following  triangle. 

Given,  the  hypotenuse  of  a  right  angled  triangle,  840.4  feet,  and  one  of 
the  oblique  angles,  38°  16',  to  find  the  other  parts. 

The   two   oblique  angles,  together,  make  90°  (th.  11,  b.  1,  cor.  4); 
therefore,  the  other  angle  is  61°  44'. 
sin.  38°  16  As  1:  38°  16'=AC  :  CB 

But  the  natural   sine  of  38°,  16'  is 
.61932  and  AC=840.4. 

Therefore,     1  :  .61932=840.4  :  CB 
840.4 


247728 
247728 
495456 

CJ?=520.476628 


PLANE    TRIGONOMETRY.  157 

For  the  side  AB,  we  have  the  following  proportion  : 

1  :cos.38o  16'=AC  :AB 

That  is,        .        .  1  :  ,78613=840.4  :  AB 

8404 


314052 
314052 
628104 
A5=659.823262 
Before  we  go  into  logarithmic  computation,  it  is  important  to  say  a 
word  or  two  in  relation  to  the  nature  of  logarithms. 

Logarithms  are  exponential  numbers ;  and  Algebra  teaches  us,  that  the 
addition  of  the  exponents  of  like  quantities  multiplies  the  quantities, 
and  the  subtraction  of  the  exponents  divides  the  quantities. 

Hence,  by  logarithms,  we  perform  miUtiplication  by  addition,  and  division 
hy  subtraction. 

EXPLANATION    OF    THE    TABLES. 

For  the  computation  of  logarithms,  we  refer  at  once  to  Algebra; 
here  we  shall  point  out  the  manner  of  finding  them  in  the  tables,  and 
some  of  their  uses.  The  logarithm  of  1,  is  0;  of  10,  is  1.00000;  of 
100,  is  2.00000,  &c.  Hence,  the  logarithm  of  any  number  between  1 
and  10,  must  be  a  decimal;  between  10  and  100,  must  be  1  and  a 
decimal;  between  100  and  1000,  must  be  2  and  a  decimal.  The  whole 
number  belonging  to  a  logarithm,  is  called  its  index.  The  index  is 
never  put  in  the  tables  (except  from  1  to  100,  and  need  not  be  put 
there),  because  we  always  know  what  it  is.  It  is  always  one  less  than 
the  number  of  digits  in  the  whole  number.  Thus,  the  number  3754 
has  3  for  the  index  to  its  logarithm,  because  the  number  consists  of 
4  digits  ;  that  is,  the  logarithm  is  3,  and  some  decimal. 

The  number  347.921  has  2  for  the  index  of  its  logarithm,  because 
the  number  is  between  347  and  348,  and  2  is  the  index  for  the  loga- 
rithms of  all  numbers  over  100,  and  less  than  1000. 

All  numbers  consisting  of  the  same  figures,  whether  integral,  frac- 
tional, or  mixed,  have  logarithms  consisting  of  the  same  decimal  part. 
The  logarithms  would  differ  only  in  their  indices. 

Thus,        .  the  number    7956.  has      3.900695  for  its  log. 

the  number    795.6  has      2.900695        " 

the  number    79.56  has       1.900695        " 

the  number    7.956  has      0.900695        " 

the  number    .7956  has —1.900695        « 

the  number  .07966  has  —2.900695        ** 


158  ELEMENTS    OF 

Prom  this  we  perceive  that  we  must  take  the  logarithm  out  of  the 
table  for  a  mixed  number  or  a  decimal,  the  same  as  if  the  figures 
expressed  an  entire  number ;  and  then,  to  prefix  the  index,  we  must 
consider  the  value  of  the  number. 

The  decimal  part  of  a  logarithm  is  always  positive  ;  but  the  index 
becomes  negative  when  the  number  is  a  decimal ;  and  the  smaller  the 
decimal,  the  greater  the  negative  index. 

To  prefix  the  index  to  a  decimal,  count  the  decimal  point  as  1,  and 
every  cipher  as  1,  up  to  the  first  significant  figure,  and  this  is  the 
negative  index. 

For  example,  find  the  logarithm  of  the  decimal  .0000831. 
Num.  0000831  log.  —6.919601 

The  point  is  counted  one,  and  each  of  the  ciphers  is  counted  one ; 
therefore  the  index  is  minus  Jive. 

The  smaller  the  decimal,  the  greater  the  negative  index  ;  and  when 
the  decimal  becomes  0,  the  logarithm  is  negatively  infinite. 

Hence,  the  logarithmic  sine  of  0°  is  negatively  infinite,  however  great 
the  radius. 

The  logarithm  of  any  number  consisting  of  four  figures,  or  less,  is 
taken  out  of  the  table  directly,  and  without  the  least  difficulty. 

Thus,  to  find  the  logarithm  of  the  number  3725,  we  find  372,  at  the 
side  of  the  table,  and  run  down  the  column  marked  6  at  the  top,  and 
we  find  opposite  the  former,  and  under  the  latter,  ,571126,  for  the  deci- 
mal part  of  the  logarithm. 

Hence,  the  logarithm  of  3725      is    3.571126 
the  logarithm  of  37250     is     4.571126 
the  logarithm  of  37.25     is     1.571126,  &c. 
Find  the  logarithm  of  the  number      834785. 

This  number  is  so  large  that  we  cannot  find  it  in  the  table,  but  we 
can  find  the  numbers  8347  and  8348.  The  logarithms  of  these  num- 
bers are  the  same  as  the  logarithms  of  the  numbers  834700  and  834800, 
except  the  indices. 

834700     log.     6.921530 
834800     log.     5.921582 

Diflference,  .        .        100  52 

Now,  our  proposed  number,  834785,  is  between  the  two  preceding 
numbers ;  and,  of  course,  its  logarithm  lies  between  the  two  preceding 
logarithms ;  and,  without  further  comment,  we  may  proportion  to  it 
thus,        .        .        .        100  :   85=52  :  44.2 

Or,      .        .        .  1.  :  .86=62  :  44.2 


PLANE    TRIGONOMETRY.  159 

To  the  logarithm         .        .        .         6.921630 
Add 44 

Hence,  the  logarithm  of   834785    is    6.921574 
the  logarithm  of  8.34785     is     0.921574 
From  this  we  draw  the  following  rule  to  find  the  log.  of  any  number 
consisting  of  more  than  four  places  of  figures. 

Rule. — TaJce  out  the  logarithm  of  the  four  superior  places,  directly  from 
the  table,  and  take  the  difference  between  this  logarithm  and  the  next  greater 
logarithm  in  the  table.     Multiply  this  difference  by  the  inferior  places  of 
figures  in  the  number,  as  a  decimal. 
Example.     Find  the  logarithm  of  357.32514. 

"    the  logarithm  of  3573.  decimal  part  is  .553033 
The  diflTerence  between  this  and  the  next  greater  in  the  table,  is  122. 
The  figures  not  included  in  the  above  logarithm,  are 

.2514 

Multiply  by        .        .        .  122 

6028 
5028 
2514 

30.6708 
This  result  shows  that  31  should  be  added  to  the  decimal  part  of  the 
logarithm  already  found  ;  that  is,  the  logarithm  of  the  proposed  number, 
357.32514    is     2.653064 
The  logarithm  of     357325.14    is    5.553064 

We  will  now  give  the  converse  of  this  problem  :  that  is,  we  give  the 
decimal  part  of  a  logarithm,  .553064,  to  find  the  figures  corresponding. 
The  next  less  logarithm  in  the  table,  is  .663033,  corresponding  to 
the  figure  3573.  The  difference  between  our  given  logarithm  and  the 
one  next  less  in  the  table,  is  31;  and  the  difference  between  two  con- 
secutive logarithms  in  this  part  of  the  table,  is  122.  Now  divide  31  by 
122,  and  write  the  quotient  after  the  number  3573. 
That  is,        .        .        .     122)31.  (254 

244 

660 
610 

500 

488 
The  figures,  then,  are  3573254,  which  corresponds  to  the  decima 
logarithm   .663064 ;   and  the  value  of  these  figures  will,  of   course, 
depend  on  the  index  to  the  logarithm. 


160  ELEMENTSOF 

Prom  this,  we  draw  the  following  rule  to  find  the  number  correspond- 
ing to  a  given  logarithm. 

Rule. — If  the  given  logarithm  is  not  in  the  table,  find  the  one  next  less, 
and  take  out  the  four  figures  corresponding;  and  if  more  than  four  figures 
are  required,  take  the  difference  between  the  given  logarithm  and  the  next  le.'^s 
in  the  table,  and  divide  that  difference  by  the  difference  of  the  two  consecutive 
logarithms  in  the  table,  the  one  less,  the  other  greater  than  the  given  loga- 
rithm; and  the  figures  arising  in  the  quotient,  as  many  as  may  be  required^ 
must  be  annexed  to  the  former  figures  taken  from  the  table. 

EXAMPLES. 

1.  Given,  the  logarithm  3.743210,  to  find  its  corresponding  number 
true  to  three  places  of  decimals.  Ans.  6636.182 

2.  Given,  the  logarithm  2.633366,  to  find  its  corresponding  number 
true  to  two  places  of  decimals.  Ans.  429.89 

3.  Given,  the  logarithm  — 3.291746,  to  find  its  corresponding 
number.  Ans.  .0019677 

TABLE    II. 

This  table  contains  logarithmic  sines  and  tangents,  and  natural  sines 
and  cosines.  We  shall  confine  our  explanations  to  the  logarithmic 
sines  and  cosines. 

The  sine  of  every  degree  and  minute  of  the  quadrant  is  given, 
directly,  in  the  table,  commencing  at  0*^,  and  extending  to  45°,  at  the 
head  of  the  table ;  and  from  46°  to  90°,  at  the  foot  of  the  table, 
increasing  backward. 

The  same  column  that  is  marked  sine,  at  the  top,  is  marked  cosine 
at  the  bottom ;  and  the  reason  for  this  is  apparent  to  any  one  who  has 
examined  the  definitions  of  sines. 

The  difference  of  two  consecutive  logarithms  is  given,  corresponding 
to  ten  seconds.  Removing  the  decimal  point  one  figure,  will  give  the 
difference  for  one  second ;  and  if  we  multiply  this  diflTerence  by  any 
proposed  number  of  seconds,  we  shall  have  a  difference  corresponding 
to  that  number  of  seconds,  above  the  logarithm,  corresponding  to  the 
preceding  degree  and  minute. 

For  example,  find  the  sine  of  19°  17'  22". 

The  sine  of  19°  17',  taken  directly  from  the  table,  is       9.618829 

The  difference  for  10"  is  60.2;  for  1",  is  6.02X22    .  133 

Hence,  19°  17'  22"  sine  is 9.618952 

Prom  this  it  will  be  perceived  that  there  is  no  difficulty  in  obtaining 
the  sine  or  tangent,  cosine  or  cotangent,  of  any  angle  greater  than  30'. 


PLANE    TRIGONOMETRY. 


161 


Conversely.  Given  the  logarithmic  sine  9.982412,  to  find  its  corres- 
ponding arc.  The  sine  next  less  in  the  table,  is  9.982404,  and  gives 
the  arc  73°  48'.  The  difference  between  this  and  the  given  sine,  is  8, 
and  the  difference  for  1",  is  .61 ;  therefore,  the  number  of  seconds  cor- 
responding to  8,  must  be  discovered  by  dividing  8  by  the  decimal  .61, 
which  gives  13.    Hence,  the  arc  sought  is  73°  48'  13". 

These  operations  are  too  obvious  to  require  a  rule.  When  the  arc 
is  very  small,  such  arcs  as  are  sometimes  required  in  astronomy,  it  is 
necessary  to  be  very  accurate ;  and  for  that  reason  we  omitted  the 
difference  for  seconds  for  all  arcs  under  30'.  Assuming  that  the  sines 
and  tangents  of  arcs  under  30'  vary  in  the  same  proportion  as  the  arcs 
themselves,  we  can  find  the  sine  or  tangent  of  any  very  small  arc  to 
great  accuracy,  as  follows ; 

The  sine  of  l',  as  expressed  in  the  table,  is  .  .  6.463726 
Divide  this  by  60  ;  that  is,  subtract  logarithm  .  .  1.778151 
The  logarithmic  sine  of  1",  therefore,  is  .  .  .  4.685575 
Now,  for  the  sine  of  17",  add  the  logarithm  of  17       .      1.230449 

Logarithmic  sine  of  17",  is 6.916024 

In  the  same  manner  we  may  find  the  sine  of  any  other  small  arc. 
For  example,  find  the  sine  of  14'  2li";  that  is,  86r'5 

To  logarithmic  sine  of  1",  is, 4.685575 

Add  logarithm  of  861.5 2.935255 

Logarithmic  sine  of  14'  21^" 7.620830 

Without  finder  preliminaries,  we  may  now  preceed  to  practical 


EXAMPLES. 


2.  In  a  right  angled  triangle,  ABC,  given 
the  base,  AB,  1214,  and  the  angle  A,  61°  40' 
30",  to  find  the  other  parts. 


To  find  BC. 


As  radius 

:    tan.A  61°  40'  30' 
::   AB  1214 
:    BC  1535.8      . 


10.000000 

10.102119 

3.084219 


3.186338 

N.  B.  When  the  first  term  of  a  logarithmic  proportion  is  radius, 
Lhe  resulting  logarithm  is  found  by  adding  the  second  and  third  loga- 
rithms, rejecting  10  in  the  index,  which  is  dividing  by  the  first  term. 

In  all  cases  we  add  the  second  and  third  logarithms  together;  which, 
in  logarithms,  is  multiplying  these  terms  together:  and  from  that  sum 
11 


162  ELEMENTS    OF 

we  subtract  the  first  logarithm,  whatever  it  may  be,  which  is  dividing 
by  the  first  term. 

To  find  AC. 

As  sin. C,  or  cos.A  61°  40'  30"  .  9.792477 

:     AB  1214  .  3.084219 

::    Radius      .  .  10.000000 

:     AC  1957.7  .  3.291742 

To  find  this  resulting  logarithm,  we  subtracted  the  first  logarithm 
from  the  second,  conceiving  its  index  to  be  13. 
Let  ABC  represent  any  plane  triangle,  right  angled  at  B. 

1.  Given  AC  73.26,  and  the  angle  A  49°  12'  20";  required  the  other 
parts  1  Ans.  The  angle  C  40°  47'  40",  BC  65.46,  and  AB  47.87. 

2.  Given  AB  469.34,  and  the  angle  A  61°  26   17",  to  find  the  other 
parts  I  Ans.  The  angle  C  38°  33'  43",  BC  588.7,  and  A  C  752.9. 

3.  Given  AB  493,  and  the  angle  C  20°  14';  required  the  remaining 
parts  1  Ans.  The  angle  A  69°  46',  BC  1338,  and  A  C  1425. 

4.  Let  AJ?=331,  the  angle  A=49°  14';  what  are  the  other  parts  ] 

Ans.  AC  506.9,  BC  383.9,  and  the  angle  C  40°  46'. 

6.  If  AC:=46,  and  the  angle  C=yi^  22',  what  are  the  remaining 
parts  I  Ans.  AB2T.Zl,BC  35.76,  and  the  angle  A  52°  38' 

6.  Given  A  C  4264.3,  and  the  angle  A  66°  29'  13",  to  find  the  remain 
ing  parts.   Ans.  AB  2354.4,  BC  3555.4,  and  the  angle  C  33°  30'  47". 

7.  If  A5=44.2,  and  the  angle  A=31°  12'  49",  what  are  the  other 
parts  1  Ans.  AC  49.35, BC  25.57,  and  the  angle  C 58°  47'  11". 

8.  If  AJ5=8372.1,  and  J5C=694.73,  what  are  the  other  parts? 

Ans.  AC  8400.9,  the  angle  C  85°  15',  and  the  angle  A  4°  45'. 

9.  If  AB  be  63.4,  and  AC  be  85.72,  what  are  the  other  parts  ] 

Ans.  BCbl.n,  the  angle  C  47°  42',  and  the  angle  A  42°  18' 

10.  Given  AC  7269,  and  AB  3162,  to  find  the  other  parts. 

Ans.  BC  6546,  the  angle  C  25°  47'  7",  and  the  angle  A  64°  12'  53". 

11.  Given  AC  4824,  and  BC  2412,  to  find  the  other  parts. 

Ans.  The  angle  A  30°  00',  the  angle  C  60°  00',  and  A5  4178 


W  or  THE 

f    UNIVERC 

\  OF 

PLANE    TRIGONOMETlMfesss^^         1G3 


OBLIQUE    AN&LED    TRIGONOMETRY. 

EXAMPLE     1. 


In  the  triangle  AJ5C,  given  AJ?=376,  the 
angle  A=48o  3',  and  the  angle  5a=40°  14', 
to  find  the  other  parts. 

As  the  sum  of  the  three  angles  of  every 
triangle  is  always  180°,  the  third  angle,  C, 
must  be  180<^--88°  l7'=9lo  43'. 

To  find  AC. 


As  8in.9lo  43' 
:  AjB376  . 
::  sin.    B  40°  14' 

:    AC 243     . 


9.999805 
2.675188 
9.810167 

12.385355 
2.385550 


Observe,  that  the  sine  of  91°  43'  is  the  same  as  the  cosine  of  1°  43'. 

To  find  BC. 
As  8in.9l°  43'  .         9.999805 

:    AB  376  .        2.575188 

::   8in.A48°  3'         .         9.871414 


BC  279.8 


12.446602 
2.446797 


EXAMPLE     2. 

In  a  plane  triangle,  given  two  sides,  and  an  angle  opposite  one  of  them, 
to  determine  the  other  parts. 

Let  AD=1751.  feet,  one  of  the 
given  sides.  The  angle  D=31°  17 
19",  and  the  side  opposite,  1257.5. 
From  these  data,  we  are  required  to 
find  the  other  side,  and  the  other  two 
angles. 

In  this  case  we  do  not  know  whether 
A  C  or  AE  represents  1257.5,  because 
AC=AE.  If  we  take  AC  for  the  other  given  side,  then  DC  is  the 
other  required  side,  and  DAC  is  the  vertical  angle.  If  we  take  AE 
for  the  other  given  side,  then  DE  is  the  required  side,  and  DAE  in  the 
vertical  angle  ;  but  in  such  cases  we  determine  both  triangles. 


164 

(Prop.  4.) 


ELEMENTS    OF 

To  find  the  angle  E=C. 

As  AC=AJE;==1257.5 

log.     3.099508 

:    D  31°  ir  19" 

sin.     9.715460 

::    AD  1751 

log.     3.243286 

12.958746 

r :    46°  18' 

sin.     9.859238 

E=C 

From  180°  take  46°  18',  and  the  remainder  is   the   angle  DCA 
=  133°  42'. 
The  angle  DAC=ACE—D  {th.  11,  b.  1);  that  is, 

DAC=46°  18'— 31°  17'  19"=15°  0'  41" 

The  angles  D  and  £,  taken  from  180°,  give  jDA£=102°  24'  41". 

To  find  DC. 

As  sin.D  31°  17'  19'      log.       9.715460 

:    AC  1257.5         .       log.       3.099508 

::   sin.Z)ACl5°0'4l"log.       9.413317 


12.512825 

:    Z>C  626.86 

2.797165 

To  find  DE. 

\s  sin.jD  31°  17'  17" 

9.715460 

:    AE  1267.5 

3.099508 

::   sin.l02°  24'  41'  . 

9.989730 

13.089238 

:    DjB  2364.7  .         .         3.373778 

N.  B.  To  make  the  triangle  possible,  AC  must  not  be  less  than 
ABi  the  sine  of  the  angle  2>,  when  DA  is  made  radius. 

EXAMPLE     3. 

In  any  plane.  triangUj  given  two  sides  and  the  included  angle,  to  find 
the  other  parts. 

Let  Aj9=1751  (see  last  figure),  Z)JEJ=2364.6,  and  the  included  angle 
Z)=31°  17'  19".  We  are  required  to  find  DE,  the  angle  DAE,  and 
angle  E.  Observe  that  the  angle  E  must  be  less  than  the  angle  DAE, 
because  it  is  opposite  a  less  side. 

From  ....         180° 

Takei?      ....  31°  17'  19" 

Sum  of  the  other  two  angles  =148°  42'  41"  (th.  11,  b.  1) 
i  sum         .        .        .        .     =  74°  21'  20" 
By  proposition  7, 


PLANE    TRIGONOMETRY.  16& 

DE+DA  :  DE^DA=:  tan.74°  21'  20 "  :  taii.i(I>A£— JB) 
That  is, 

4116.6  :  613.6=  tan.740  21'  20"  :  i(DAE—E) 
Tan.74°  21'  20"         .        10.662778 
613.6  .         .  2.787816 

13.340693 
4116.6  log.  (sub.)  3.614423 

i{DAE—E)  tan.28o  1'  36"     9.726170 
But  the  half  sum  and  half  difference  of  any  two  quantities  are  equal 
to  the  greater  of  the  two  ;  and  the  half  sum,  less  the  half  difference, 
is  equal  the  less. 

Therefore,  to    74°  21'  20" 
Add        .  28       1    36 


DA£=102°  22' 

66" 

E=  46 

19 

44 

Tojind  AE. 

As  sin.E  A^  19'  44" 

, 

9.869323 

:   DA  1761      . 

. 

3.243286 

::  sin.2>  31°  17'  19" 

• 

9.715460 
12.968746 

:   AE  1267.2  . 

• 

3.099423 

EXAMPLE    4. 

Given  the  three  sides  of  a  plane  triangle  to  find  the  angles. 
Given  AC=1761,  05=1267.6,  AjB=2364.6 
If  we  take  the  formula  for  cosines,  we  will 
compute  the  greatest  angle,  which  is  C.    To 
correspond  with  the  formula. 


cos.iC=i/?!£^fiZ£j   we  must 
V  ah 

take     a— 1257.6  5=1761,  and  c=2364.6 
The  half  sum  of  these  is,      5=2686.6  •  «— c=322 
2^2        .        .     20.000000 
5=2686.6      .       3.429187 
5— c=322  .      2.607866 

Numerator,  log.      26.937043 


166  ELEMENTSOF 

R^        .         .     20.000000 
5=2686.5      .       3.429187 
s— c=322  .       2.507856 

Numerator,  log.    25.937043 
a  1257.6     3.099508 
h  1751.       3.243286 
Denominator,  log.  6.342794      6.342794 
2)19.594249 
iC=  51°  ir  10"  COS.  9.797124 
C=102    22  20 
The  remaining  angles  may  now  be  found  by  problem  4. 

We  give  the  following  examples  for  practical  exercises : 
Let  ABC  represent  any  oblique  angled  triangle. 

1.  Given  AB  697,  the  angle  A  81°  30'  10",  and  the  angle  B  40° 
80'  44",  to  find  the  other  parts. 

Ans,  AC  534,  BC  813,  and  the  angle  C  57°  59'  6". 

2.  If  AC=720.8,  the  angle  A=70°  5'  22",  and  the  angle  5=59* 
35'  36",  required  the  other  parts. 

Ans.  AB  643.2,  BC  785.8,  and  the  angle  C  50°  19'  2". 

3.  Given  BC  980.1,  the  angle  A  7°    6'  26",  and  the  angle  B  106* 
2'  23",  to  find  the  other  parts. 

Ans.  AB  7284,  AC  7613.3,  and  the  angle  C  66°  61'  11". 

4.  Given  AB  896.2,  BC  328.4,  and  the  angle  C  113°  45'  20",  to 
find  the  other  parts. 

Ans.  AC  712,  the  angle  A  19°  36'  48",  and  the  angle  B  46°  38'  62". 

6.  Given  AC  4627,  BC  6169,  and  the  angle  A  70°  26'  12",  to  find 
the  other  parts. 
Ans.  AB  4328,  the  angle  B  57°  29'  66",  and  the  angle  C  52°  4'  52". 

6.  Given  AB  793.8,  BC  481.6,  and  AC  600.0,  to  find  the  angles. 
Ans.  The  angle  A  35°  16'  32",  the  angle  B  36°  49'  18",  and  the 

angle  C  107°  66'  10". 

7.  Given  AB  100.3,  BC  100.3,  and  AC  100.3,  to  find  the  angles. 

Ans.  The  angle  A  60°,  the  angle  B  60°,  and  the  angle  C  60°. 

8.  Given  AB  92.6,  BC  46.3,  and  AC  71.2,  to  find  the  angles. 
Ans.  The  angle  A  29°  17'  22",  the  angle  B  48°  47'  31",  and  the 

angle  C  101°  66'  8". 


PLANE    TRIGONOMETRY.  167 

9.  Given  AB  4963,  BC  6124,  and  AC  6621,  to  find  the  angles. 
Ans,  The  angle  A  67°  30'  28",  the  angle  B  67°  42'  36",  and  the 
angle  C  64°  46'  66". 

10.  Given  AB  728.1,  BC  614.7,  and  AC  683.8,  to  find  the  angles. 
Ans.  The  angle  A  64°  32'  62",  the  angle  B  60°  40'  68",  and  the 

angle  C  74°  46'  10". 

11.  Given  AB  96.74,  BC  83.29,  and  AC  111.42,  to  find  the  angles. 
Ans.  The  angle  A  46°  30'  46",  the  ang*e  B  76°  3'  46",  and  the  angle 

C  67°  25'  30' . 

12.  Given  AB  363.4,  BC  148.4,  and  the  angle  JB  102°  18'  27",  to 
find  the  other  parts. 

Atw.  The  angle  A  20°  9' 17",  the    side    AC.  =  420  8,     and  the 
angle  C  67°  32'  16". 

13.  Given  AB  632,  BC  494,  and  the  angle  A  20°  16',  to  find  the 
other  parts,  C  being  acute. 

Ans.  The  angle  C  26°  18'  19",  the  angle  B  133°  26'  41",  aud  AC 
1035.86. 

14.  Given  AB  63.9,  AC  46    21 ,  and  the  angle  B  68916,' to  find  the 
other  parts. 

Ans.  The  angle  A  38°  68',  the  angle  C  82°  46,  and  BC  34,16. 

16.  Given  AB  2163,  BC  1672,  and  the  angle  C  112°  18'  22",  to 
,  find  the  other  parts. 

Ans.  AC  877.2,  the  angle  B  22°  2'  16",  and  the  angle  A  45°  39'  22". 

16.  Given  AB  496,  BC  496,  and  the  angle  B  38°  16',  to  find  the 
other  parts. 

Ans.  AC  326.1,  the  angle  A  70°  62'  and  the  angle  C70°  62'. 

17.  Given  AB  428,  the  angle  C  49°  16',  and  (AC+jBC)  918,  to 
find  the  other  parts,  the  angle  JS  being  obtuse. 

Ans.  The  angle  A  38°  44'  48",  the  angle  B  91°  59'  12",  AC  664.49, 
and  BC  353.5. 

18.  Given  AC  126,  the  angle  B  29°  46',  and  {AB—BC)  43, to  find 
the  other  parts. 

Ans.  The  angle  A  66°  61'  32",  the  angle  C  94°  22'  28",  A5  253.05, 
and  BC  210^.54. 

19.  Given  AB  1269,  AC  1837,  and  the  angle  A  63°  16'  20",  to  find 
the  other  parts. 

Ans.  The  angle  B  83°  23'  47",  the  angle  C  43°  19'  63",  and  BC 
1482.16. 


168  EXE  ME  NTS    OF 

APPLICATION    OF    TRIGONOMETRY    TO    MEA- 
SURING  THE    EIGHT    AND    DISTANCES   OF 
VISIBLE    OBJECTS. 

In  this  useful  application  of  trigonometry,  a  base  line  is  always  sup- 
posed to  be  measured,  or  given  in  length  ;  and  by  means  of  a  quadrant, 
sextant,  circle,  theodolite,  or  some  other  instrument  for  measuring 
angles,  such  angles  are  measured  as  connected  with  the  base  line, 
and  the  objects  whose  bights  or  distances  it  is  proposed  to  determine, 
enable  us  to  compute,  from  the  principles  of  trigonometry,  what  those 
hights  or  distances  are. 

Sometimes,  particularly  in  marine  surveying,  horizontal  angles  are 
determined  by  the  compass;  but  the  varying  effect  of  surrounding 
bodies  on  the  needle,  even  in  situations  little  removed  from  each  other, 
and  the  general  construction  of  the  instrument  itself,  render  it  unfit  to 
be  applied  in  the  determination  of  angles  where  anything  like  precision 
is  required. 

The  following  examples  present  sufficient  variety  to  guide  the  student 
in  determining  what  will  be  the  most  eligible  mode  of  proceeding  in 
any  case  that  is  likely  to  occur  in  practice. 

EXAMPLE     1. 

Being  desirous  of  finding  the  distance  between  two  distant  objects, 
C  and  X>,  I  measured  a  base  ABj  of  384  yards,  on  the  same  horizontal 
plane  with  the  objects  C  and  D.  At  A,  I  found  the  angle  Z)AJS=48° 
12',  and  CAJ5=890  18';  at  B  the  angle  ABC  was  46°  14',  and  ABD 
87°  4'.  It  is  required  from  these  data  to  compute  the  distance  between 
C  and  D. 

From  the  angle  CAB,  take  the  angle  DAB;  the 
remainder,  41°  6',  is  the  angle  CAD.  To  the  angle 
DBA,  add  the  angle  DAB,  and  44°  44',  the  supple- 
ment of  the  sum,  is  the  angle  ADB,  In  the  same 
way  the  angle  A  CB,  which  is  the  supplement  of 
the  sum  of  CAB  and  CBA,  is  found  to  be  44°  28'. 
Hence,  in  the  triangles  ABC  and  ABD,  we 
have 

As  sin.  A  CB  44°  28  .        9.846405 

:    AB  384  yards       .         .        2.684331 
::  sin.  ABC  46°  14'  .        9.858635 

12.442996 


AC  395.9  yards    .        .        2.597561 


PLANE    TRIGONOMETRY. 


109 


As  sin.  ADB  44*^  44' 

9.847454 

:    AB  384  yards 

2.584331 

::   sin.  ABD  87°  4'  . 

9.999431 

12.583762 


:    AD  544.9  yards    .         .         2.7363.08 
Then,  in  the  triangle  CAD,  we  have  given  the  sides  CA  and  AD, 

and  the  included   angle  CAD,  to   find  CD;  to  compute  virhich  we 

proceed  thus : 

The  supplement  of  the  angle  CAD  is  the  sum  of  the  angles  A  CD, 

and  ADC; 

Hence,     .    it =69°  27',  and,  by  proportion  we  have. 


2 

AsAD+AC 
:  AD^AC     . 

::  tan.  ^^^+^^^ 
2 

940.8 
149 

69°  27' 
22    54 

2.937497 
2.173186 

10.426108 

.    .        A  CD— ADC 

:    tan.  

2 

12.599294 
9.625797 

the  angle  A  CD     sum 
the  angle  ADC  .  difF. 

As  sin. ADC  46°  33' 
:    A  C  395.9  yards  . 
::  sin.  CAD  41°  6' 

:    CD  358.5  yards  . 

92    21 
46    33 

9.860922 
2.597585 
9.817813 
12.415398 
2.554476 

EXAMPLE     2. 

To  determine  the  altitude  of  a  lighthouse,  I  observed  the  elevation  of 
its  top  above  the  level  sand  on  the  seashore,  to  be  15°  32'  18",  and 
measuring  directly  from  it,  along  the  sand  638  yards,  I  then  found  its 
elevation  to  be  9°  56'  26";  required  the  hight  of  the  lighthouse. 

Let  CD  represent  the  hight  of  the  lighthouse 
above  the  level  of  the  sand,  and  let  B  be  the 
first  station,  and  A  the  second  ;  then  the  angle 
CBD  is  15°  32'  18",  and  the  angle  CAB 
is  9°  56'  26";  therefore,  the  angle  ACB,  which 
is  the  difference  of  the  angles  CBD  and  CAB, 
is  5°  35'  52". 


170 

ELEMENTS 

OF 

Hence, 

.    As  sin. ACB  6°  36' 52"     . 

8.989201 

:    AB  6Z8 

2.804821 

::  sin.  angle  A  9°  56' 26" 

9.237 1U7 
12.041928 

:   BC  1129.06  yards 

3.052727 

As  radius  .... 

10.000000 

:    BC  1129.06 

3.052727 

::  sin.CjBD  15°  32'  18"  . 

9.427945 
12.480672 

:    DC  302.46  yards 

2.480672 

EXAMPLE 

3. 

Coming  from  sea,  at  the  point  D,l  observed  two  neadlands.  A  and  B, 
and  inland,  at  C,  a  steeple,  which  appeared  between  the  headlands. 
I  found,  from  a  map,  that  the  headlands  were  5.35  from  each  other ; 
that  the  distance  from  A  to  the  steeple  was  2.8  miles,  and  from  B 
to  the  steeple  3.47  miles  ;  and  I  found  with  a  sextant,  that  the  angle 
ADC  was  12°  15',  and  the  angle  BDC  15°  30'.  Required  my  distance 
from  each  of  the  headlands,  and  from  the  steeple. 


CONSTRUCTION. 

The  angle  between  the  two  headlands  is  the 
sum  of  150  30'  and  12°  15',  or  27°  45'.  Take 
the  double,  55°  30'.  Conceive  AB  to  be  the 
chord  of  a  circle,  and  the  segment  on  one  side 
of  it  to  be  55°  30  ;  and,  of  course,  the  other  will 
be  304°  30'.  The  point  D  will  be  somewhere  in 
the  circumference  of  this  circle.  Consider  that 
point  as  determined,  and  join  CD. 

In  the  triangle  ABC  we  have  all  the  sides,  and,  of  course,  we  can 
find  all  the  angles  ;  and  if  the  angle  ACB  is  less  than  (180°— (27° 
45')  )=152°  15',  then  the  circle  cuts  the  line  CD,  in  a  point  E,  and 
C  is  without  the  circle. 

Join  AE,  BE,  AD,  and  DB.  AEBD  is  a  quadrilateral  in  a  circle, 
and  AEB-^ADB=180°. 

The  angle  ADE=  the  angle  ABE,  because  both  are  measured  by 
half  the  arc  AE.     Also,  EDB=EAB,  for  a  similar  reason. 

Now,  in  the  triangle  AEB,  its  side  AB,  and  all  its  angles,  are 
known ;  and  from  thence  AE  can  be  computed.    Then,  having  the 


PLANE    TRIGONOMETRY. 


171 


two  sides  AC  and  AE  of  the  triangle  AEC,  and  the  included  angle 
CAE,  we  can  find  the  angle  AEC,  and,  of  course,  its  supplement, 
AED.     Then,  in  the  triangle  AED  we  have  the  side  AE,  and  the 
two  angles  AED  and  ADE,  from  which  we  can  find  AD. 
The  computation,  at  length,  is  as  follows  : 

To  find  AE. 

16°  30'  As  sin.AEB  162°  15'      .    9.668027 

12    15  :    A5  5.35      .         .         .       .728354 

::    sin. ABE     12°  15'      •     9.326700 


angle  EAB 
angle  EBA 


27   45 
180      0 


angle  AEB  152    15 


AE  2.438 


To  find  the  angle  BA  C. 
BC    3.47 

AB    5.35    log.      .728354 
AC    2.80    log.      .447158 


2)11.62 


BC 


5.81 
2.34 


log. 
log. 


1.175512 

.764176 
.369216 
20 


17°  41'  58" 

2 

angle 
angle 

BAC   35    23  56 
EAB    15    30 

angle 

EAC   19    53  56 

180 

2)160      6     4 

80      3     2 

22^133392 

2)19.957880 

cos.  9.978940 


AEC-^ACE 
2 


10.355054 
.387027 


To  find  the  angles  AEC  and  ACE. 
As  AC-\-AE  6.238  .719165 

:    AC—AE  .362 

AEC^ACE 


tan.- 


—80°  3'  2' 


AEC— ACE 
tan. 7i 21  30  12 


-1.558709 

10.755928 

10.314637 
9.596472 


172 

ELEMENTS 

OF 

angle 

AEC 

ACE  01 

ACD 

101*: 

>33'14' 

'  sum 

angle 

68 

32  50 

diff. 

angle 

CDA 

12 

15 

70 

47  50 

supplement  1 

35    23  56    angle  CAB 


73    48  14 

To  find  AD. 

Assin.ADC  12°  15'       . 

9.326700 

:    AC  2.8     . 

.447158 

::   sin.  A  CD  58°  32' 50" 

9.930985 

10.378143 

:     AD  11.26  miles 

1.051443 

EXAMPLE     4. 

The  elevation  of  a  spire  at  one  station  was  23°  50'  17",  and  the 
horizontal  angle  at  this  station,  between  the  spire  and  another  station, 
was  93°  4'  20".  The  horizontal  angle  at  the  latter  station,  between 
the  spire  and  the  first  station,  was  54°  28'  36",  and  the  distance  between 
the  two  stations,  416  feet.     Required  the  hight  of  the  spire. 

Let  CD  be  the  spire,  A  the  first  station,  and  B 
the  second ;  then  the  vertical  angle  CAD  is  23° 
60'  17";  and  as  the  horizontal  angles  CAB  and 
CJ5A  are  93°  4'  20",  and  54°  28'  36"  respectively, 
the  angle  A  CB,  the  supplement  of  their  sum,  is 
32°  27'  4". 

To  find  AC, 


As  sin.  5  CA  32°  2T  3" 
:    side  AB  416 
::   sin.AJ5C  54°  28'  36' 


9.729634 

2.619093 

9.910560 

12.529653 


:  side  AC  631 

2.800019 

To  find  DC. 

\.s  radius    .... 

10.000000 

:    side  AC  631 

2.800019 

::   tan.DAC  23°  60'  17"   . 

9.645270 

:    DC  278.8 

2.445289 

PLANE    TRIGONOMETRY.  173 

By  the  application  of  the  fourth 
example  we  can  compute  the  dif- 
ferent elevations  of  different  planes, 
provided  the  same  object  is  visible 
from  them. 

For  example,  let  M  be  a  promi- 
nent tree  or  rock  near  the  top  of  a 
mountain,  and  by  observations  taken 
at  Ay  we  can  determine  the  perpendicular  Mn.  By  like  observations 
we  can  determine  the  perpendicular  Mm.  The  difference  between 
these  two  perpendiculars,  is  nm,  or  the  difference  in  the  elevation 
between  the  two  points  A  and  B.  But  if  the  distances  between  A 
and  n,  or  B  and  m,  are  considerable,  or  more  than  two  or  three  miles, 
corrections  must  be  made  for  the  convexity  of  the  earth  ;  but  for  less 
distances  such  corrections  are  not  necessary. 

EXAMPLES      FOR      EXERCISE. 

1.  Required  the  hight  of  a  wall  whose  angle  of  elevation  is  observed, 
jit  the  distance  of  463  feet,  to  be  16°  21'  1  A7is.   135.8  feet. 

2.  The  angle  of  elevation  of  a  hill  is,  near  its  bottom,  31°  18',  and 
214  yards  further  off,  26°  18'.  Required  the  perpendicular  hight  of  the 
hill,  and  the  distance  of  the  perpendicular  from  the  first  station. 

Ans.  The  hight  of  the  hill  is  665.2,  and  the  distance  of  the  perpen- 
dicular from  the  first  station,  is  929.6. 

3.  The  wall  of  a  tower  which  is  149.6  feet  in  hight,  makes,  with  a  line 
drawn  from  the  top  of  it  to  a  distant  object  on  the  horizontal  plane,  an 
angle  of  57°  21'.  What  is  the  distance  of  the  object  from  the  bottom 
of  the  tower  1  Ans.  233.3  feet. 

4.  From  the  top  of  a  tower,  whose  hight  was  138  feet,  I  took  the 
angles  of  depression  of  two  objects  which  stood  in  a  direct  line  from 
the  bottom  of  the  tower,  and  upon  the  same  horizontal  plane  with  it. 
The  depression  of  the  nearer  object  was  found  to  be  48°  10',  and 
that  of  the  further,  18°  62'.  What  was  the  distance  of  each  from  the 
bottom  of  the  tower  1 

Ans.  Distance  of  the  nearer  123.6,  and  of  the  farther  403.8  feet. 
6.  Being  on  the  side  of  a  river,  and  wishing  to  know  the  distance 
of  a  house  on  the  other  side,  I  measured  312  yards  in  a  right  line  by 
the  side  of  the  river,  and  then  found  that  the  two  angles,  one  at  each 
end  of  this  line,  subtended  by  the  other  end  and  the  house,  were  31^ 
15'  and  86°  27'.  What  was  the  distance  between  each  end  of  the  line 
and  the  house  ?  Ans.  351.7,  and  182.8  yards. 


174  ELEMENTSOF 

6.  Having  measured  a  base  of  260  yards  in  a  straight  line,  close  by 
one  side  of  a  river,  I  found  that  the  two  angles,  one  at  each  end  of 
the  line,  subtended  by  the  other  end  and  a  tree  close  to  the  opposite 
bank,  were  40°  and  80°.    What  was  the  breadth  of  the  river  'i 

Ans.  190.1  yards. 

7.  From  an  eminence  of  268  feet  in  perpendicular  bight,  the  angle 
of  depression  of  the  top  of  a  steeple  which  stood  on  the  same  hori- 
zontal plane,  was  found  to  be  40°  3',  and  of  the  bottom  66°  18'.  What 
was  the  bight  of  the  steeple  ?  Ans.  117.8  feet. 

8.  Wanting  to  know  the  distance  between  two  objects  which  were 
separated  by  a  morass,  I  measured  the  distance  from  each  to  a  point 
where  I  could  see  them  both  ;  the  distances  were  1840  and  1428  yards, 
and  the  angle  which,  at  that  point,  the  objects  subtended,  was  36°  18' 
24".     Required  their  distance.  Ans.  1090.85  yards. 

9.  From  the  top  of  a  mountain,  three  miles  in  hight,  the  visible  hori- 
zon appeared  depressed  2°  13'  27".  Required  the  diameter  of  the  earth, 
and  the  distance  of  the  boundary  of  the  visible  horizon. 

Ans.  Diameter  of  the  earth  7958  miles,  distance  of  the  horizon 
154.54  miles. 

lOr  From  a  ship  a  headland,  was  seen  bearing  north,  39°  23'  east. 
After  sailing  20  miles  north,  47°  49'  west,  the  same  headland  was 
observed  to  bear  north,  87°  11'  east.  Required  the  distance  of  the 
headland  from  the  ship  at  each  station '] 

Ans.  The  distance  at  the  first  station  was  19.09,  and  at  the  second 
26.96  miles. 

11.  The  top  of  a  tower,  100  feet  above  the  level  of  the  sea,  was 
seen  as  on  the  surface  of  the  sea,  from  the  masthead  of  a  ship,  90  feet 
above  the  water.  The  diameter  of  the  earth  being  7960  miles,  what 
was  the  distance  between  the  observer  and  the  object  1 

Ans.  23.9  plus  -j^  for  refraction  =  25.7  miles. 

12.  From  the  top  of  a  tower,  by  the  seaside,  of  143  feet  high,  it  was 
observed  that  the  angle  of  depression  of  a  ship's  bottom,  then  at  anchor, 
measured  35°;  what,  then,  was  the  ship's  distance  from  the  bottom  of 
the  wall  1  Ans.  204.22  feet. 

1 3.  Wanting  to  know  the  breadth  of  a  river,  I  measured  a  base  of 
600  yards  in  a  straight  line  close  by  one  side  of  it ;  and  at  each  end  of 
this  line  I  found  the  angles  subtended  by  the  other  end  and  a  tree  close 
to  the  bank  on  the  other  side  of  the  river,  to  be  63°  and  79°  12'.  What 
then,  was  the  perpendicular  breadth  of  the  river  1     Ans.  629.48  yards. 

14.  What  is  the  perpendicular  hight  of  a  hill,  its  angle  of  elevation 
taken  at  the  bottom  of  it,  being  46°,  and  200  yards  further  ofi*,  on  a 
level  with  the  bottom,  the  angle  was  31°1  Am.  286.28  yards. 


PLANE    TRIGONOMETRY.  175 

15.  Wanting  to  know  the  hight  of  an  inaccessible  tower ;  at  the 
least  distance  from  it,  on  the  same  horizontal  plane,  I  took  its  angle  of 
elevation  equal  to  68°;  then  going  300  feet  directly  from  it,  found  the 
angle  there  to  be  only  32°;  required  its  hight,  and  my  distance  from 
it  at  the  first  station.  .        }  Hight       307.53. 

^^'  ^Distance  192.15. 

16.  Two  ships  of  war,  intending  to  cannonade  a  fort,  are,  by  the  shal- 
lowness of  the  water,  kept  so  far  from  it,  that  they  suspect  their  guns 
cannot  reach  it  with  eiSect.  In  order,  therefore,  to  measure  the  dis- 
tance, they  separate  from  each  other  a  quarter  of  a  mile,  or  440  yards, 
then  each  ship  observes  and  measures  the  angle  which  the  other  ship 
and  fort  subtends,  which  angles  are  83°  45'  and  85°  15'.  What,  then, 
is  the  distance  between  each  ship  and  the  fort  1    .        S  2292.26        , 

^^^-   J  2298.05  y^'^"- 

17.  A  point  of  land  was  observed  by  a  ship,  at  sea,  to  bear  east-by- 
south  ;*  and  after  sailing  north-east  12  miles,  it  was  found  to  bear  south- 
east-by-east. It  is  required  to  determine  the  place  of  that  headland, 
and  the  ship's  distance  from  it  at  the  last  observation  1 

Ans.  26.0728  miles. 

18.  Wanting  to  know  my  distance  from  an  inaccessible  object,  0,  on 

the  other  side  of  a  river  ;  and  having  no  instrument  for  taking  angles, 

but  only  a  chain  or  chord  for  measuring  distances  ;  from  each  of  two 

stations,  A  and  JB,  which  were  taken  at  500  yards  asunder,  I  measured 

in  a  direct  line  from  the  object  0,  100  yards,  viz.,  AC  and  BD,  each 

equal  to  100  yards ;  also  the  diagonal  AD  measured  550  yards,  and 

the  diagonal  BC  560.     What,  then,  was  the  distance  of  the  object  0 

from  each  station  A  and  jBI  A'»      i  ^^  536.25. 

^^-   I  BO  500.09. 

19.  A  navigator  found,  by  observation,  that  the  vertex  of  a  certain 
mountain,  which  he  supposed  to  be  45  minutes  of  a  degree  distant,  had 
an  altitude  above  the  sea  horison  of  31'  20".  Now,  on  the  supposition 
that  the  earth's  radius  is  3956  miles,  and  the  observer's  dip  was  4'  15", 
what  was  the  hight  of  the  mountain  I  Ans.  3960  feet. 

N.  B.  This  should  be  diminished  by  about  its  one-eleventh  part  foi 
the  influence  of  horizontal  refraction. 

•  That  is,  one  point  south  of  east.    A  point  of  the  compass  is  11°  15. 


176  .  SPHERICAL 


SPHERICAL     TRIGONOMETRY. 

Spherical  Geometry  is  nothing  more  than  the  general  principles 
of  geometry  applied  to  the  various  sections  of  a  sphere  ;  and  spher- 
ical trigonometry,  is  but  the  general  principles  of  plane  trigonome- 
try applied  to  triangles  resting  on  a  surface  of  a  sphere,  and  the 
planes  of  the  sides  of  the  triangles  passing  through  the  center  of 
the  sphere. 

DEFINITIONS. 

1.  A  sphere  is  a  solid  whose  surface  is  equally  convex  in  every 
part,  and  every  point  of  the  surface  is  equally  distant  from  one  point 
within,  and  this  point  is  called  the  center.  A  sphere  may  be  con- 
ceived to  be  generated  by  the  revolution  of  a  semicircle  about  its 
diameter. 

If  the  center  of  the  semicircle  rests  at  the  same  point,  the  posi- 
tion of  the  diameter  may  be  in  any  direction  or  position,  and  the 
revolution  of  the  semicircle  will  describe  the  same  sphere. 

2.  Any  plane  that  passes  through  the  center  of  the  sphere,  di- 
vides the  solid  and  the  surface  into  two  equal  parts. 

3.  Any  two  planes  that  pass  through  the  center  of  a  sphere,  in- 
tersect each  other  on  the  opposite  points  of  the  sphere,  because  the 
section  of  any  two  planes  is  a  right  line  (th.  2,  b.  6). 

4.  A  great  circle  on  a  sphere,  is  one  whose  plane  passes  through 
the  center  of  the  sphere. 

5.  Every  great  circle  has  poles,  two  points  on  the  sphere  directly 
opposite  to  each  other  and  equally  distant  from  every  point  on  the 
great  circle. 

The  distance  from  any  pole  to  its  equator  in  a»y  direction,  is  one 
fourth  of  the  whole  distance  round  the  sphere. 

6.  Any  point  on  a  sphere  may  be  a  pole  to  some  ff recti  circle. 

7.  A  spherical  triangle  is  formed  by  the  intersection  of  three 
great  circles  on  a  sphere.  Conceive  three  radii  drawn  from  the 
three  angular  points  to  the  center  of  the  sphere,  thence  forming  a 
solid  angle.  The  angles  of  the  three  planes  which  form  this  solid 
angle  at  the  center,  are  the  three  angles  which  measure  the  sides 
of  the  triangle,  and  the  inclination  of  these  planes  to  each  other 
form  the  angles  of  the  triangle. 


TRIGONOMETRY.  177 

8.  The  complete  measure  of  a  spherical  triangle,  is  but  the 
complete  measure  of  a  solid  angle  at  the  center  of  a  sphere  ;  and 
this  solid  angle  is  the  same,  whatever  be  the  radius  of  the  sphere. 

9.  Every  great  circle,  or  portion  of  a  great  circle  on  the  surface 
of  a  sphere,  has  its  poles ;  conversely,  every  pole,  or  the  point 
where  two  circles  intersect,  has  its  equator  90°  distant,  and  the 
portion  of  this  equator  between  the  two  sides,  or  the  two  sides 
produced,  measures  the  spherical  angle  at  the  pole. 

The  inclination  of  two  tangents  of  two  arcs  formed  at  their  point 
of  intersection,  also  measures  the  spherical  angle.    (Def.  5,  to  b.  6). 

10.  We  can  always  draw  one,  and  only  one  great  circle  through 
any  two  points  on  the  surface  of  a  sphere  ;  for  the  two  given  points 
and  the  center  of  the  sphere,  give  three  points,  and  through  three 
points  only  one  plane  can  be  made  to  pass  (cor.  th.  1,  b.  6). 

PROPOSITION   1. 

Every  section  of  a  sphere  hy  a  plane  is  a  circle. 

If  the  plane  passes  through  the  center  of  the  sphere,  the  section 
is  evidently  a  circle,  for  every  point  on  the  surface  of  the  sphere  is 
equally  distant  from  the  center.  These  sections  are  great  circles, 
and  all  great  circles  on  the  same  sphere  are  equal  to  each  other. 

Now  let  the  cutting  plane  not  pass  through  the  center.  From 
the  center  (7,  let  fall  Cn  perpendicular 
to  the  plane  ;  and  when  a  line  is  perpen- 
dicular to  a  plane,  it  is  perpendicular 
to  all  Hues  that  can  be  drawn  in  that 
plane  (th.  3,  b.  6);  therefore,  any  line 
as  nm  in  the  plane,  is  at  right  angles  to 
Cn.     Hence  nm=  J  Cm^ — Cn^. 

But  nm  is  any  line  in  the  plane,  from  the  point  n  to  the  surface 
of  the  sphere,  and  this  value  for  nm  is  invariable,  and  it  is  the 
radius  of  a  circle  whose  center  is  n, 

N.  B.  These  circles  are  called  small  circles,  and  are  greater  or 
less,  as  they  are  nearer  or  more  remote  from  the  center  C 

Small  circles  on  a  sphere,  are  never  considered  as  sides  of  spheri- 
cal triangles.  We  again  repeat,  that  sides  of  spherical  triangles 
must  be  portions  of  ^reat  circles,  and  each  side  must  be  less 

than  180°. 

12 


178 


PHERICAL 


PROPOSITION     2. 

Any  two  sides  of  a  spherical  triangle  are  together  greater  than  the 
third. 

Let  A  By  AC,  and  BO,  be  the  three  sides  of 
the  triangle,  and  D  the  center  of  the  sphere. 

The  arcs  AB,  AO,  and  BC,  are  measured 
by  the  angles  of  the  planes  that  form  the  solid 
angle  at  J).  But  any  two  of  these  angles  are 
together  greater  than  the  third  (th.  10,  b.  6). 
Therefore,  any  two  sides  of  the  triangle  are  together,  greater  than 
the  third.     Q.  K  D. 

PROPOSITION     3. 

The  sum  of  ths  three  sides  of  any  spherical  triangle  is  less  than  the 
circumference  of  a  great  circle. 

Let  ABO  be  a  triangle ;  the  two  sides 
AB,  AO,  produced,  will  meet  at  the  point 
on  the  sphere  which  is  directly  opposite 
to  A;  and  the  arcs  ABD,  and  A  OD,  are 
together  equal  to  a  great  circle.  But  by 
the  last  proposition,  BO  is  less  than  the 
two  arcs  BJ)  and  DO.  Therefore,  AB,  BO,  and  A (7,  are  together 
less  than  ABD-^-A  OD;  that  is,  less  than  a  great  circle.     Q.  E.  D. 


PROPOSITION     4. 

Every  right  angled  spherical  triangle  must  have  a  complemental, 
supplemental,  and  four  quadrarUal  triangles  in  the  same  hemisphere. 

Let  ABO,  be  a  right  angled  spherical 
triangle,  right  angled  at  B. 

Produce  the  sides  AB  and  AO,  and 
they  will  meet  at  A',  the  opposite  point  on 
the  sphere.  Produce  BO,  both  ways,  90° 
from  the  point  B,  to  P  and  P',  which  are 
therefore,  poles  to  the  arc  AB  (def.  9, 
spherics).  Through  A,  P,  and  the  center  of  the  sphere,  pass  a 
plane  cutting  the  sphere  into  two  equal  parts,  forming  a  great  circle 
on  the  sphere,  which  great  circle  will  be  represented  by  the  plane 


TRIGONOMETRY.  179 

circle  PAP' A  on  tlie  paper.  At  right  angles  to  this  plane,  pass 
another  plane,  cutting  the  sphere  into  two  equal  parts  ;  this  great 
circle  is  represented  on  the  paper,  by  the  straight  line  P  OP'.  A 
and  A'y  are  the  poles  to  the  great  circle  P  OP',  P  and  P',  are  the 
poles  to  the  great  circle  ABA', 

As  PCy  PD  and  CD,  are  portions  of  great  circles  on  a  sphere, 
CPD  is  a  spherical  triangle,  and  it  is  crnnplemental  to  the  given 
triangle  ABC;  because  CD  is  the  complement  of  AC,  CP  the 
complement  of  BC,  and  PD  is  the  complement  of  D  0,  or  of  the  an- 
gle A.  Again,  the  triangle  A'BC,  is  supplemental  to  ABC,  because 
A'=A;  A' Ch  the  supplement  of  AG,  and  A'B  is  the  supplement 
of  AB.  ACP  is  a  spherical  triangle,  and  one  of  its  sides,  AP,  is 
a  quadrant,  and  it  is  therefore  called  a  quadrantal  triangle.  So  also, 
are  the  triangles  A'CP,  ACP',  and  P'CA',  quadrantal  triangles. 

Cor.  In  every  triangle  there  are  six  elements ;  three  sides  and 
three  angles,  which  are  sometimes  called  parts. 

Now,  if  all  the  parts  of  the  triangle  ABC  are  known,  the  parts 
of  the  complemental  triangle  PCD,  are  also  known,  and  the  sup- 
plemental triangle  A' BC,  must  be  as  completely  known. 

When  the  triangle  PCD  is  known,  the  triangles  A  CP  and  A' PC 
are  also  known,  for  the  side  PD,  measures  the  angles  PA  C  and 
PA  C,  and  the  angle  CPD,  added  to  the  right  angle  A' PD,  gives 
the  angle  A' PC,  and  CPA,  is  supplemental  to  this.  Hence  a 
solution  of  any  right  angled  spherical  triangle,  is  a  solution  to  its 
complemental,  supplemental,  and  all  its  quadrantal  triangles. 

Definition.  Every  triangle,  together  with  its  supplemental  tri- 
angle, form  what  is  called  a  Lune.  Thus,  the  triangles  ABC  and 
A'BC,  form  a  lune  ;  PCD  and  P'CD,  form  a  lune  ;  PAC  and 
P'A  C,  also  form  a  lune. 

It  is  obvious,  that  the  surface  of  the  lune  PAP'B,  is  to  the 
surface  of  the  sphere,  as  the  arc  AB,  is  to  the  wliole  circumference. 

PROPOSITION     5. 

If  there  he  three  arcs  of  great  circles  whose  poles  are  the  angular 
points  of  a  sj^herical  triangle,  such  arcs,  if  produced,  will  form, 
another  triangle,  whose  sides  will  he  supplemental  to  the  angles  of  the 
first  triangle,  and  the  sides  of  the  first  tnangle  will  he  supplemental  to 
the  angles  of  the  second 


180  SPHERICAL 

Let  the  arcs  of  the  three  great  circles  be 

QHy  PQf  KLy  whose  poles  are  respectively  ,4, 

Bt  and  0.     Produce  the  three  arcs  until  they 

meet  in  E^  i>,  and  F,     We  are  now  to  show, 

that  E  is  the  pole  to  the  great  circle  AG;  D 

the  pole  of  the  great  circle  BC;  F  the  pole  to 

the  great  circle  AB,     Also,  that  the  side  EF^ 

is  supplemental  to  the  angle  A;  ED  to  the 

angle  C;  and  DF  to  the  angle  B;  and  also, 

that,  the  side  A  (7,  is  supplemental  to  the  angle  E^  (fee. 

Any  pole  is  90°  from  any  point  on  its  great  circle,  and  therefore, 

as  A  is  the  pale  to  the  great  circle  GHy  the  point  Ai  is  90°  from 

the  point  E.    As  Q  is  the  pole  of  the  great  circle  LKy  C  is  90°  from 

any  point  in  that  great  circle ;  therefore,  G  is  90°  from  the  point 

Ey  and  Ey  being  90°  from  both  A  and  G,  it  is  the  pole  of  the  arc 

A  G.     In  the  same  manner,  we  may  prove  that  D  is  the  pole  of 

BGy  and  i^the  pole  of  AB, 

Because  A  is  the  pole  of  the  arc  GHy  the  arc  OH  measures  the 

angle  A  (def.  9  spherics);  for  the  same  reason,  PQ  measures  the 

angle  By  and  XJT  measures  the  angle  G. 

Because  ^is  the  pole  of  the  arc  AGy  E  11=90° 
Or,         ...         .  EG-\-OJI=90° 

For  a  like  reason,    .         .  FH-\-GH=90° 

Adding  these  two  equations,  and  obseiTing  that  GH=^Ay  and 
afterward  transposing  one  Ay  we  have, 

EG^GH^FH=\ZO°--A. 

Or, EF=\ZO°—A     ^ 

In  like  manner,        ....    i^i>=180°— i?      I    (a) 

And, jE'i>=180°— a     J 

But  the  arc  (180° — ^),  is  a  supplemental  arc  to  A,  by  the  de- 
finition of  arcs  ;  therefore,  the  three  sides  of  the  triangle  EDF,  are 
supplements  of  the  angles  Ay  By  (7,  of  the  triangle  ABC. 

Again,  as  Ey  is  the  pole  of  the  arc  A  C,  the  whole  angle  E,  is 
measured  by  the  whole  arc  LIT. 

But,       ....  AG-\-GB=90° 

Also,     ....  AG-\-AL=90° 

By  addition,  .        .  A  G+A  (7-f  Cir-\-AL=  1 80° 


TRIGONOMETRY.  181 

By  transposition,  .  A  C+  CII+AL=  1 80°—^  C 

That  is,   ....   Zff,  or  JS^ISO^-^AO  ^ 

In  the  same  manner,         .        .       F^ISO^'—AB  \    (b) 

And, J9=180°— ^(7  J 

That  is,  the  sides  of  the  first  triangle,  are  supplemental  to  the 
angles  of  the  second  triangle.     Q.  E,  D. 

PROPOSI  TION     6. 

The  sum  of  the  three  angles  of  any  spherical  triangle,  is  greater 
than  two  right  angles,  and  less  than  six  right  angles. 

Turn  to  equations  (a),  of  the  last  proposition,  and  add  them  to- 
gether. The  first  member  of  the  equation  so  formed  will  be  the 
sum  of  three  sides  of  a  spherical  triangle,  which  sum  we  may  des- 
ignate by  aS'.  The  other  member  will  be  6  right  angles  (there 
being  2  right  angles  in  each  180°)  less  the  three  angles  A,  B, 
and  0. 

That  is,         .         .    iSr=6  right  angles— (^+^+(7) 

By  proposition  3,  the  sum  S,  is  less  than  4  right  angles;  there- 
fore, to  it  add  s,  a  sufiicient  quantity  to  make  4  right  angles. 

Then,      4  right  angles=6  right  angles — (A-{-B-{-C)-\-s 
Drop  4  right  angles  from  both  members,  and  transpose  (A-\-B-\-  C) 

Then,         .  A-\-B-]-C=2  right  angles+5 

That  is,  the  three  angles  of  a  spherical  triangle,  make  a  greater 
sum  than  two  right  angles  by  the  indefinite  quantity  s,  which  quan- 
tity is  called  the  spherical  excess,  and  is  greater  or  less  according  to 
the  size  of  the  triangle. 

Again  the  sum  of  the  angles  is  less  than  6  right  angles.  There 
are  but  three  angles  to  any  triangle,  and  no  one  of  them  can  come 
up  to  180°,  or  2  right  angles.  For  an  angle  is  the  inclination  of 
two  lines  or  two  planes  ;  and  when  two  planes  incline  by  1 80°,  the 
planes  are  parallel,  or  are  in  one  and  the  same  plane  ;  therefore,  as 
neither  angle  can  equal  2  right  angles,  the  three  can  never  equal 
6  right  angles.     Q.  K  D. 

Scholium.  By  merely  inspecting  the  figure  to  proposition  4,  we 
perceive  that  the  triangle  PAB,  has  two  right  angles ;  one  at  A, 
the  other  at  B,  besides  the  third  angle  APB. 

The  triangle  P'A'O,  has  3  right  angles.  The  triangle  A'P'Q^ 
has  two  of  its  angles,  each  greater  than  a  right  angle. 


182  SPHERICAL 

PROPOSITION.     7. 

Wkh  the  sines  of  the  sides,  and  the  tangent  of  one  side  of  any 
right  angled  spherical  triangle,  two  plane  triangles  can  he  f(yrmed  thai 
will  be  similar,  and  similarly  situated. 

Let  ABO,  be  a  spherical  triangle,  right 
angled  at  JB;  and  let  D  be  the  center  of  the 
sphere.  Because  the  angle  CBA,  is  a  right 
angle,  the  plane  CDB,  is  perpendicular  to  the 
plane  DBA.  From  0,  let  fall  CII,  perpendic- 
ular to  the  plane  DBA,  and  as  the  plane  CBD 
is  perpendicular  to  the  plane  DBA,  QH  will 
lie  in  the  plane  CBD,  and  be  perpendicular 
to  the  line  DB,  and  perpendicular  to  all  lines  that  can  be  drawn  in 
the  plane  DBA,  from  the  point  ^(th.  3,  b.  6). 

Draw  HO-  perpendicular  to  DA,  and  join  GC;  GC  will  lie 
wholly  in  the  plane  CD  A  (def.  of  planes),  and  CHG  is  a  right 
angled  triangle,  right  angled  at  R. 

We  mil  now  demonstrate  that  the  angle  DGC,  is  a  right  angle. 
The  right  angled  A  CHG,  gives  CH''-^HG''=  GG''  ( 1 ) 

The  right  angled  A  DGH,  gives  DG''-\-HG''=DFP  (2) 

By  subtraction,         .         .         .    CH''—DG''=^CG'—DIP{^) 
By  transposition,       .         .         .     CH^-{-DH''=CG''^DG^'\) 
But  the  first  member  of  the  equation  (4),  is  equal  to  CD^;  be- 
cause CDH,  is  a  right  angled  triangle ; 

Therefore, CD'^^^GC-VDG'' 

Hence,  CD,  is  the  hypotenuse  to  the  right  angled  triangle  DGC 
(th.  36,  b.  1). 

From  the  point  B,  draw  BE  at  right  angles  to  DA,  and  BF  at 
right  angles  to  DB,  in  the  plane  CDB  extended  ;  the  point  F  being 
in  the  line  DC.  Join  EF,  and  as  F  is  in  the  plane  CD  A,  and  E  is 
in  the  same  plane,  the  line  EF,  is  in  the  plane  CD  A.  Now  we  are 
to  show,  that  the  triangle  CHG  is  similar,  and  similarly  situated  to  the 
triangle  BEF. 

As  HG  and  BE  are  both  at  right  angles  to  DA,  they  are  paral- 
lel ;  and  as  GH  and  BF  are  both  at  right  angles  to  DB,  they  are 
parallel;  and  by  reason  of  the  parallels,  the  angles  GHG  and 
EBF,  are  equal ;  but  GEO  is  a  right  angle ;  therefore,  EBF  is 
also  a  right  angle. 


TRIGONOMETRY.  183 

Now  as  QH  and  BE  are  parallel,  and  CH  and  BF  parallel,  we 
have,  DH '.  DB=HG  :  BE 

And,        .        .        .       DH:DB=HC'.BF 
Therefore,         .         .       HO  :  BE=^HC  :  BF  (th.  6,  b.  2) 
Or,  .        ,        ,       HQ  '.  HC=BE  :  BF 

Here,  then,  are  two  triangles,  having  an  angle  in  the  one  equal  to 
an  angle  in  the  other,  and  the  sides  about  the  equal  angles  pro- 
portional ;  the  two  triangles  are  therefore  equiangular  (th.  20,  b.  2); 
and  they  are  similarly  situated,  for  their  sides  make  equal  angles 
at  H  and  B  with  the  same  line,  DB.     Q.  E.  D. 

Scholium.  By  the  definition  of  sines,  cosines,  and  tangents,  we 
perceive,  that  CH  is  the  sine  of  the  arc  BOy  DH  is  its  cosine,  and 
BF  its  tangent ;  CO  is  the  sine  of  the  arc  A  C,  and  D  0  its  cosine. 
Also,  BE  is  the  sine  of  the  arc  AB,  and  DE  is  the  cosine  of  the 
same  arc.  With  this  figure  we  are  prepared  to  demonstrate  the 
following  theorems. 

PROPOSITION    8.      THEOREM     1. 

In  any  right  angled  spherical  triangle^  the  sine  of  one  side  is  to  the 
tangent  of  the  other  side,  as  radius  is  to  the  tangent  of  the  angle 
adjacent  to  the  first-mentioned  side. 

Or,  as  the  sine  of  one  side  is  to  the  tangent  of  the  other  side,  so  is 
the  cotangent  of  the  angle,  adjacent  to  the  first-mentioned  side,  to  the 
raditis. 

In  the  right  angled  plane  triangle  EBF,  we  have, 
EB  :  BF=E  :  tan.BEF 

That  is,         .      sin.  c  :  tan.a=B  :  tan.-4  Q.  E.  D, 

A  modification  of  this  proposition  demonstrates  the  latter  part  of 

the  theorem.     By  reference  to  equation  (5),  plane  trigonometry, 

B^ 

we  shall  find  that,  tan.-4.  cot.A=^JR^;  therefore,  tan.^= ■- 

cot.A 

Substituting  this  value  for  tangent  A,  in  the  preceding  proposi- 
tion, and  dividing  the  last  couplet  by  B,  we  shall  have. 

sm.  c  :  tan.a=  1  : r 

cot.-d 

Or,        .        .    sin.  c  :  tan.a=cot.-4  :  B  Q,  E,  J), 

Or,        .        .        .   jB  sin.  c  =tan.a  cot.^  (1) 


184 


SPHERICAL 


Cor.  By  changing  the  construction,  drawing  the  tangent  to  A  B, 
in  place  of  the  tangent  to  BC,  and  proceeding  in  a  similar  manner, 
we  have,  i2  sin.a=tan.c  cot.  (7        (2) 

PROPOSITION    9.     THEOREM.     2. 

In  any  right  angled  spherical  triangle,  the  sine  of  the  right  angle  is 
to  the  sine  of  the  hypotemise,  as  the  sine  of  either  of  the  other  angles  to 
tlie  sine  of  the  side  opposite  to  that  angle. 

N.  B.  For  the  sake  of  perspicuity,  if  not  of  brevity,  we  will  repre- 
sent the  angles  of  the  triangle,  by  A,  By  C,  and  of  the  sides  or  arcs 
opposite  to  these  angles  by  a,  b,  c;  that  is,  a  opposite  A,  &,c. 

The  sine  of  90°,  or  radius,  is  designated  by  R. 

In  the  plane  triangle  CHO,  we  have, 

sm.CIIO:  CO=:sm.Oair:  CH 
That  is,         .         .     E  :  sin.5=sin.^  :  sin.a  Q.  JS,  D, 

Or,       .         .         .      '  i2sin.a=sm.d  sin.^         (3) 
Cor.     By  a  change  in  the  construction  of  the  figure,  drawing  a 

tangent  to  AB,  &c.,  we  shall  have, 

H  :  sin.  5=sin.  C :  sin.c  Q.  E.  D. 

Or,       .         .         .         i2sin.c=sin.5  sin.  (7        (4) 
Scholium,     Collecting  the  four  preceding  equations  drawn  from 

theorems  1  and  2,  we  have, 

( 1 )  a  sin.c=tan.a  cot.^ 

(2)  i2  8in.a=tan.c  cot.  (7 

(3)  i2sin.a=sin.5sin.^ 

(4)  JR  sin.c=sin.6  sin.  C 

These  equations  refer  to  the  right  angled 
triangle  ABC;  but  the  principles  are  true 
for  any  right  angled  spherical  triangle. 
Let  us  apply  them  to  the  right  angled  tri- 
angle  PDC,   the    complemental  triangle 

to^^a 

this  application,  equation  (1)  becomes, 

R  sin.  CD=iQii.PD  cot.  C  (n) 

(2)  becomes      i2  sin.Pi>=tan.  (7i>  cot.P  {m) 

(3)  becomes      i2sin.Pi)=sin.P(7sin.C7  (o) 

(4)  becomes      i2sin.CZ>=sin.P(7sin.P  (p) 


TRIGONOMETRY.  185 

By  observing  that       sin.  CD=cos.A  C=cos.b, 

And  that  .  .  tan.Pi>=cot.i)  0=cotAy  &c ;  and  by 
running  equations  (»),  (m),  (o),  and  (p),  back  into  the  triangle 
ABCf  and  we  shall  have, 

(6)     Rcos.h=coi.A  cot.  07- 

(6)  i2cos.u4=cot.5  tan.c 

(7)  i2cos.^=cos.a  sin.(7 

(8)  R  cos.5=cos.a  cos.c 
By  observing  equation  (6),  we  find  that  the  second  member 

refers  to  sides  adjacent  to  the  angle  A.     The  same  relation  holds 
in  respect  to  the  angle  C,  and  gives, 

(9)  R  cos.C=cot.i  tan.a 
Making  the  same  observations  on  (7),  we  infer, 

(10)     i2  cos.(7=cos.c  sin.^ 

Observation  1 .  Several  of  these  equations  can  be  deduced  geo- 
metrically without  the  least  difficulty.  For  example,  take  the  fig- 
ure to  proposition  7.  Observe  the  parallels  in  the  plane  DBAf 
which  give,  DB  :  DH=DE  :  DO 

That  is,         .         .  R  \  cos.a=cos.c  :  cos.5 

'  A  result  identical  with  equation  (8),  and  in  words  is  expressed 
thus  :  As  radius  is  to  cosine  of  one  side,  so  is  the  cosine  of  the  other 
side,  to  the  cosine  of  the  hypotenuse. 

Observation  2.  Equations  numbered  from  (1)  to  (10),  cover 
every  possible  case  that  can  occur  in  right  angled  spherical  trig- 
onometry, but  the  combinations  are  too  various  to  be  remembered, 
and  readily  applied  to  practical  use. 

We  can  remedy  this  inconvenience,  by  taking  the  complement  of 
the  hypotenuse,  and  the  complements  oi  the  two  oblique  angles,  in 
place  of  the  arcs  themselves. 

Thus  h  is  the  hypotenuse,  and  let  h'  be  its  complement. 

Then,     5-f-5'=90° ;      or,   5=90°— 5';     and,     sin.5=cos.6', 

cos.6=sin.6';     tan.5=cot.5'.     In  the  same  manner  if -4' 
is  the  complement  to  A, 

Then,  .  sin.^=cos.^V  cos.-4=sin.J[V  and,  tan.-4=cot.-4V 
and  similarly,  sin.  C=  cos.  CV   cos.  (7=  sin.  C",    and  tan.  (7=  cot.  C. 


186  SPHERIC  A  I, 

Substituting  these  values  for  J,  A,  and  C,  in  the  foregoing  ten 
equations  (a  and  c  remaining  the  same),  we  have, 

Napier's     circular  parts. 

U)     Ji  sin.c=tan.a  tan.^'  I         Omitting  the  consid- 

12)  i2sin.a=tan.ctan.(7'  i     eratlon  of  the  right  augle 
^e*\       n    '                    jf            J,  i     there    are    five   parts. — 

13)  Jt  sm.a= COS. 0  cos.A  i    r,    ,         .   .  , 
'  iLach   part    taken    as    a 

14)  i2sin.C=COS.5'cOS.(7'  i     middle  part,  is  connect- 

15)  jRsin.6'=tan.^' tan.C"  ed  to  its  adjacent  parts 

16)  i?  sin.^'=tan.6'  tan.C  by     one    equation,    and 

17)  i2  sin.^'  =  COS.a  cos.  (7'  ^o  its  extreme  parts  by 
io\      D    •     r/                                                 another   equation;    and 

18)  ic  sin.o=cos.a  cos.c  i     .      „ 
(  therefore,  ten  equations 

19)  i?sm.(7'=tan.5'tan.«  are  required  for  the  com- 

20)  jR  sin.  (7'=C0S.C  COS.^' '  binations  of  all  the  parts. 

These  equations  are  very  remarkable,  because  the  first  members 
are  all  composed  of  radius  into  some  sine,  and  the  second  members 
are  all  composed  of  the  product  of  two  tangents,  or  two  cosines. 

To  condense  these  equations  in'o  words,  for  the  purpose  of  assist- 
ing the  memory,  we  will  refer  them,  any  one  of  them,  directly  to 
the  right  angled  triangle  ABC,  in  the  last  figure. 

When  the  right  angle  is  left  out  of  the  question,  a  right  angled 
triangle  consists  oi  five  parts — three  sides,  and  two  angles.  Let  any 
one  of  these  parts  be  called  a  middle  j)art,  then  two  otlier  parts 
will  lie  adjacent  to  this  part,  and  two  opposite  to  it,  that  is,  separated 
from  it  by  two  other  parts. 

For  instance,  take  equation  (11),  and  call  c  the  middle  part,  then 
A'  and  a  will  be  adjacent  parts,  and  C  and  h'  opposite  parts. 
Again,  take  a  as  d^  middle  part,  then  c  and  C'will  be  adjacent  parts, 
and  A'  and  b'  will  be  opposite  parts  ;  and  thus  we  may  go  round 
the  triangle. 

Take  any  equation  from  (11)  to  (20),  and  consider  the  middle 
part  in  the  first  member  of  the  equation,  and  we  shall  find  that 
they  correspond  to  these  two  invariable  and  comprehensive  rules. 

1 .  The  radius  into  the  sine  of  the  middle  part  equals  the  product 
of  the  tangents  of  the  adjacent  parts, 

2.  The  radius  into  the  sine  of  the  middle  part  equals  the  product  of 
the  cosines  of  the  opposite  parts. 


TRIGONOMETRY.  187 

These  rules  are  known  as  Napier's  Rules,  because  they  were  first 
brought  forth  by  that  distinguished  mathematician,  who  was  also 
the  inventor  of  logarithms. 

We  caution  the  pupil  to  be  very  particular  to  take  the  complements 
of  the  hypotenuse,  and  the  complements  of  the  oblique  angles. 


OBLIQUE    ANGLED    SPHERICAL 
TRIGONOMETRY. 

The  preceding  investigations  have  had  reference  to  right  angled 
spherical  trigonometry  only;  but  the  application  of  these  prin- 
ciples cover  oblique  angled  trigonometry  also,  for  every  oblique 
angled  spherical  triangle  may  be  considered  as  made  up  of  the 
sum  or  difference  of  two  right  angled  spherical  triangles.  With 
this  explanatory  remark,  we  give, 

PROPOSITION    9.    THEOREM.     3. 

In  all  spherical  triangles,  the  sines  of  the  sides  are  to  each  other ,  as 
the  sines  of  the  angles  opposite  to  them. 

This  was  proved  in  relation  to  right  angled  triangles  in  theorem 
2,  and  we  now  apply  the  principle  to  oblique  angled  triangles. 

Let  ABC,  be  the  triangle,  and  let  CD 
be  perpendicular  to  AB,  or  to  AB  pro- 
duced as  represented  in  the  margin. 

Then  by  theorem  2,  we  have, 

R  :  sin.^  (7=sin.  J  :  sin.  CD 

Also,    .  sm.CB  :  i2=sin.  CD:  sm.B. 

By  multiplying  these  two  proportions 
term  by  term,  and  leaving  out  the  com- 
mon factor  jR,  in  the  first  couplet,  and  the 
common  factor  sin.  CD,  in  the  second,  we 
have,  sm.CB  :  sin.^  C=sin.^  :  sin.5.     Q.  E.  D. 

C(yr,  From  the  truth  of  this  theorum,  it  follows,  that  the  angles 
at  the  base  of  an  isosceles  triangle  are  equal,  and  that  in  every 
spherical  triangle  the  greater  angle  is  opposite  the  greater  side. 


188  SPHERICAL 

PROPOSITION    10.    THEOREM     I. 
In  any  spherical  triangle,  if  an  arc  he  let  fall  frma  any  angle  to 
the  opi^osUe  side  aa  a  hose,  or  to  the  base  produced,  the  cosines  of  the 
other  two  sides  mil  be  to  each  other  as  the  cosines  of  the  segments  of 
the  base. 

By  the  application  of  equation  (8)  to  the  last  figure,  we  have, 

R  COS. A C=s cos, AD  cos. DC 
Similarly,         .        R  cos.BC^=cos.DC  cos.BD 

Dividing  one  of  these  equations  by  the  other,  omitting  common 
factors  in  numeratprs  and  denominators,  we  have, 

cos.^  C COS.  AD 

cos.BO^cos.BD 

Or,        .       cos.^C  :  cos.BO=cos.AD  :  cos.BD.     Q.  E.  D, 

PROPOSITION!!.     THEOREM    5. 

J^  from  any  angle  of  a  spherical  triangle,  a  perpendictdar  he  let 
fall  on  the  base,  or  on  the  base  produced,  the  tangents  of  the  segments 
of  the  base  will  be  to  each  other  reciprocally  proportional  to  the  cotan- 
gents of  the  segments  of  the  angle. 

By  the  application  of  equation  (2)  to  the  last  figure,  we  have, 

R  sm.CD=iB.n,AD  cot. A  CD 
Similarly,  .  R  sm.CD=i^xi.BD  cot.BCD 
Therefore,  by  equality, 

tan.^Z>  cot.J[(7i?=tan.^i>  cot.BCD 
Or,        .      tan.^i>  :  t&ji.BD=cot.BCD  :  cot.ACD.  Q.  E.  D, 

PROPOSITION     !2.     THEOREM     6. 

The  same  construction  remaining ,  the  cosines  of  the  angles  cU  the 
extremities  of  the  segments  of  the  base,  are  to  each  other  as  the  sines  of 
the  segments  of  the  opposite  angle. 

Equation  (7)  applied  to  the  triangle  A  CD,  gives 

R  cos.^=cos.  CD  sin.A  CD    (s) 

Also,        .         .        i2cos.^=cos.Ci>sin.-B(7i>    (t) 


TRIGONOMETRY.  189 

Dividing  equation  (s)  by  (t),  gives 

cos.^ sin.^  CJ) 

cos^~smlBa5 
Or,    .        .        cos.i?  :  cos.^=sin.J5  C?i>  :sin.^(7i).    Q.  JS,  D, 

PROPOSITION    13.     THEOREM    7. 

The  same  constniciion  remaining,  the  sines  of  the  segments  of  the 
dose,  are  to  each  other  as  the  cotangents  of  the  adjacent  angles. 

Equation  ( 1 ),  applied  to  the  triangle  A  CD,  gives 

B  Bm.AD=  tan.  CD  cot. A  (s) 

Similarly,         .       B  sm.£D=  tan.OZ)  cot.^  (t) 

Dividing  (s)  by  (t),  gives 

sin.^J9     cot.-4 

mnJBD~''cotJB 

Or,        .       8m.BD  :  sin.^i>=cot.5  :  cot.-4.     Q.  E.  D, 

PROPOSITION     14.     THEOREM    8. 

The  same  construction  remaining,  the  cotangents  of  the  two  sides  are 
to  each  other  as  the  cosineb  of  the  segments  of  the  angle. 

Equation  (9),  applied  to  the  triangle  ACD,  gives 

B  COS.A  CD=  Qoi.A  C  tan.  CD  {a) 

Similarly,    .        i2  cos.^(7i>=cot.J5(7tan.C7i>  {t) 

Dividing  («)  by  {t),  gives 

coB.ACD  _coi.AC 
cos.BGD'^cotBC 

Or,         .     cot  AC  :  cot.J5C7=  cos.^C7i>  :  cos.B  CD.    Q.  E.  D, 

Remark.  The  preceding  theorems  enable  us  to  solve  any 
spherical  triangle,  right  angled  or  oblique,  when  any  three  of  the 
*M?  parts  are  given.  But  oblique  angled  spherical  triangles  we  have 
thus  far  considered  as  composed  of  two  right  angled  triangles ; 
and  it  is  sometimes  a  little  troublesome  to  select  the  theorems  or 
equations  which  apply  to  the  case  in  question.    To  remedy  this 


190  SPHERICAL 

inconvenience,  we  will  at  once  seek  a  relation  between  the  cosines 
and  sines  of  an  angle  of  any  spherical  triangle,  and  the  sines  and 
cosines  of  its  sides.  Therefore,  we  investigate  the  following 
propositions. 

PROPOSITION    15.     PROBLEM. 

Investigate,  snd  show  the  relation  between  the  cos  we  of  an  angle  of 
a  spherical  triangle,  and  the  sines  and  cosines  of  its  sides. 

Let  ABC  be  a  spherical  triangle,  and 
CD  a  perpendicular  from  the  angle  C 
on  to  the  side  AB,  or  on  to  the  side  AB 
produced.    Then,  by  proposition  10,  th.  4, 

cos.^C  :  COS.  (7-5=cos.^i>  :  cos.BD  (1) 

When  CD  falls  within  the  triangle, 

BD=(AB—AD) 
When  CD  falls  without  the  triangle, 

BD=(AD—AB) 
Hence,         .  cos.-Bi>=cos.(-4i) — AB) 

Now,  cos.(-4^ — AD)=cos.(AD — AB),   because  each  of  them 
is  equal  to  cos. AB  cos.-4i>4-sin.^jB  sin. AD.     (Plane  trig.  eq.  10.) 

This  value  of  cos.BD,  put  in  proportion  (1),  gives 
cos.AC  :  COS. OB=^cos. AD  :  cos.-4^  cos.^2)-j-sin.^^  sin.^i)  (2) 

Dividing  the  last  couplet  of  proportion  (2)  bycos.^J9,  observing 

that    .        .        .       — ^7-=r=tan.-4i>,  and  we  have 
cos.AD 

cos.AC  :  cos.CB=l  :  cos.^^+sin.^^  tan. AD     (3) 

By  applying  equation  (6)  to  the  triangle  A  CD,  taking  the  radius 

as  unity,  we  have  cos.-4=cot.-4C7tan.^i>  (k) 

But,         .  tan.-4Ccot.^(7=l     (eq.  5,  plane  trig.)  (I) 

Multiply  equation  (k)  hy  tan. AC,  observing  equation  (/),  and 
we  have      .     tan.^(7cos.^=tan.^2) 

Substituting  this  value  of  tan. AD,  in  proportion  (3),  we  have 
cos.  .4  C  :  cos.  CB=  1  :  cos.-4.5+  sin.-4jB  tan.-4  C  cos. A        (  4  ) 


TRIGONOMETRY.  191 

Multiplying  extremes  and  means,  gives 
COS.  CB=coa.A  C  cos.^-S+sin.^jB(cos.^  C  tan.  A  C)co8.A 

But,        .        .  tan. AC=: — '-t-?^*  or,  cos.^(7tan.^(7=sin.J[(7 
cos.^  u 

Therefore,        .   cos.(75=cos.^(7cos.-4^+sin.-4^sin.-4(7cos.-4 

„                        .     cos.CB — cos.^C7cos.^5  ,_^.  ^    ,         .^ 
Hence,    .     cos.-4= :: — --jr—. — -—: (F)  final  result.* 

By  processes  perfectly  similar,  like  theorems  may  be  deduced  for 
the  angles  B  and  C. 

If  the  sides  opposite  the  angles  A,  B,  and  0,  be  respectively 
represented  by  a,  b,  and  c,  the  formula  will  be  expressed  thus  : 

COS. a — cos.5  cos.c^ 


cos.^= 


cos.jB= 


sin.6  sin.c 
C0S.5 — cos. a  cos.c 


COS.  C- 


sm.a  sm.c 
COS.C — COS. a  C0S.5 


sin.a  sin.6 


{S) 


*  As  this  equation  has  been  denominated"  The  fundamental  formida 
of  Spherical  Trigonometry,''^  and  as  it  is  susceptible  of  a  more  geome- 
trical demonstration,  we  give  the  following,  which  we  beUeve  will  be 
very  acceptable  to  every  lover  of  mathematical  science. 

Let  ABC  be  a  spherical  triangle,  and 
O  the  center  of  the  sphere. 

From  the  angle  A,  draw  AD  tangent 
to  the  arc  AB,  and  AE  tangent  to  the 
arc  A  C.  OD  and  OE,  drawn  from  the 
center  of  the  sphere  to  the  extremities  of 
the  tangents,  are,  of  course,  secants.  OD 
is  the  secant  of  AB,  and  OE  the  secant  of  the  arc  AC. 

Because  AD  is  a  tangent,  it  is  perpendicular  to  the  radius  OA.  For 
the  same  reason  AE  is  perpendicular  to  the  same  radius  OA.  But 
OA  is  the  common  intersection  of  the  two  planes  A 0J5  and  AOCy 
and  hence,  by  definition  5,  book  6,  the  angle  DAE  is  the  inclination 
of  the  two  planes  AOB  and  AOC,  and  is,  therefore,  equal  to  the 
spherical  angle  A.  As  is  customary,  let  the  side  opposite  to  A  be 
designated  by  a,  opposite  B  by  b,  opposite  C  by  c. 


192  •  SPHERICAL 

These  formulas  are  not  adapted  to  the  use  of  logarithms ;  and 
the  use  of  natural  sines  and  cosines  would  lead  to  tedious  operations; 
we  must,  therefore,  make  some  advantageous  mutations,  or  the 
equations  will  be  useless  ;  hence  the  following  investigations  : 

.    In  equation  (35),  plane  trigonometry,  we  find 

1 + cos  .-4 = 2cos^^^ 

COS. a — cos.h  cos.c 


Therefore,        2  cos.2|-4=  1  -f 


sin.b  sin.  c 


(sm.b  sin.c — cos.b  cos.c)4-cos.a  , 

—i r—^- '- (m) 

sm.6  sm.c  ^ 

But,  .  cos.(^-f-c)=cos.J  cos.c — sin.c  sm.b  (9),  plane 
trigonometry.  By  comparing  this  last  equation  with  the  second 
member  of  equation  (m),  we  perceive  that  equation  (m)  is  readily 
reduced  to 

cos. a — cos(5-|-c) 

^  sm.b  sm.c 


Then,      AD=  tan.c,  AE=  tan.6,  0D=  sec.c,  0E=  sec.b. 

Designate  DE  by  x,  and  observe  that  the  angle  BOC  is  measured 
by  the  arc  JBC=a. 

Now,  to  the  two  plane  triangles  ODE  and  ADE,  if  we  apply  equa- 
tion (?n),  proposition  8,  plane  trigonometry,  we  shall  have 

sec.2  c-l-sec.2  b — x^ 

cos.a= ! 

2  sec.^  sec.o 

.      tan.2c4-tan.2  6 — x^ 

cos.A= ■— — 

2  tan.c  tan.o 

Clearing  these  two  equations  of  fractions,  and  subtracting  the  latter 
from  the  former,  and  observing,  that  for  any  arc,  sec.^ — tan.^=R~ ;  and 
if  R  is  unity,  as  it  is  in  this  case,  we  shall  have, 

2  sec.c  sec.Z>  cos. a — 2  tan.c  tan.&  cos.A=2 

Dividing  by  2,  and  substituting  the  values  of  the  secants  and  tan- 
gents from  equations  (4)  and  (5),  plane  trigonometry, 

Namely,        .         sec.=  — ,    tan.=^* ,    we  shall  then  have, 
cos.  cos. 

cos. a  sin.c  sin.&  cos.A  ^  , 


<X>S.C    cos.b  cos.c  C0S.6 


TRIGONOMETRY.  198 

Considering  (b+c)  as  one  arc,  and  then  making  application  of 
equation  (18),  plane  trigonometry,  we  have, 

2  C0S^:i--4  = ; r— : 

^  sm.o  sm.c 

But,        .  — - — = — a;  and  if  we  put  S  to  rep- 

resent    — - — ,  we  shall  have 

^A     sm,S  sm.(S — a) 

cos^— = r-^-4 

2  sm.o  sm.c 


^  ^       Isin.S  sm.(S — a) 

Or,  .        .      COS.— =-J .    ,   : 

2      ^        sm.6  sm.c 

The  right  hand  member  of  this  equation  gives  the  value  of  the 

Clearing  of  fractions,  transposing,  and  changing  signs,  will  give 

BUi.o  sin.&  cos.A=cos.a — cos.<7  cos.& 

_,       .  .     co%.a — cos.^  cos.ft 

Therefore,        .        .        .      cos.A= ; r-i 

8m.6^  sm.o 

For  the  sake  of  the  mathematical  exercise,  I  will  suppose  we  have 
the  three  sides  of  a  spherical  triangle,  as  follows : 

a=70°  4'  18",  fe=59°  16'  23",  and  c=63°  21' 27",  from  which  we 
require  the  angle  A,  and  we  have  no  other  formula  except  the  above 
equation,  amA  logarithms  are  not  yet  invented. 

From  the  table  of  natural  sines  and  cosines,  we  find 
cos.«=0.34090 

cos.6=0.61l91     sin.5=0.8791 
cos.c=0.44840     sin.c=:0.8938 
By  the  multiplication  of  decimals,  retaining  only  five  places,  we  find, 
COS.&  cos.c=0.22953,  and  sin.&  sin.c=0.76786 

From  cos.o        .         .         0.34890 
Take  cos.6  cos.c         .         0.22953 

0.76786)0. 1 1 137(0. 14506=cos.  A 

By  comparing  this  decimal  with  the  table,  we  find  it  very  nearly 
corresponds  to  81°  40'.     The  true  value  of  A  is  81°  38'  20" 
13 


194 


SPHERICAL 


cosine  when  the  radius  is  unity.  To  a  greater  radius,  the  cosine 
would  be  greater  ;  and  in  just  the  same  proportion  as  the  radius 
increases,  all  the  trigonometrical  lines  increase  ;  therefore,  to  adapt 
the  above  equation  to  our  tables  where  the  radius  is  R,  we  must 
write  R  in  the  second  member,  as  a  factor;  and  if  we  put  it  und  r 
the  radical  sign,  we  must  write  R"^. 

For  the  other  angles  we  shall  have  precisely  similar  equations ; 


That  is 


A        IRhm.Ssin.iS—a) 
COS.— =x/ r    ^    / 


COS 


B_  jRhm.S  sm.{  S—b) 


|-^/ 


sm.a  sm.c 


Rhin.S  sm.{S—c) 
sin.a  sin.b 


{T) 


Formulas,  for  the  sines  of  the  angles,  are  obtained  as  follows : 
From  equation  (32),  plane  trigonometry,  we  obtain 

2  sin.^^^=l — C0S.-4. 
Substituting  the  value  of  cos.-4,  taken  from  equation  {S)^  and 

COS. a — cos. 6  cos.c 


we  have 


2  sin.2i^=l. 


sin.5  sin.c 
(sin.5  sin.c+cos.fi  cos.c) — cos.a 
sin. 6  sin.c 
But,  cos.(ficrc)=sin.J.sin.c-|-cos.6  cos.c     (  (10)  plane  trig.) 
This  equation  reduces  the  preceding  one  to 

cos.(5cr  c) — cos.a 
sin .6  sin.c 


2sm.2|^=- 


Considering  {h<j>  c)  as  a  single  arc,  and  applying  equation  ( 1 8), 
plane  trigonometry,  we  have 

^    .     /a+5 — c\    .     (  a-\-c — b\ 

2  sm?^A=^ 


But, 
Also, 


a-\-h — c__a-|-5+c 
2      ~       2      " 

2  2 


sin.6  sin.c 


c=  S — c,  if  we  put  S- 


a-\-b-{'C 
:      _ 


TRIGONOMETRY. 


195 


Dividing  the  preceding  equation  by  2,  and  making  these  sub- 
stitutions, we  have, 

,   .     sm.(S — c)sin.(AS^ — h)       ,  j*      •        -x 

[.\A= ^   .    /  . — ^ -f  when  radius  is  unity. 


Sin. 


sin.6  sm.c 
When  radius  is  i2,  we  have 


sm.^A=^j 


^  sin.(/S— c)sin.(/S— 6) 
sin.5  sin.c 


«.    .1    ,         .    ,  T,       /i22sin.(>S'--a)sin.(AS^— c) 
Similarly,    sin4^=^ —      ^         ^ ^         ^ 


sin.a  sin.c 


And, 


'^       ^  sin.a  sin.6 


sin 


(«7) 


To  apply  to  our  tables,  H?  must  be  put  under  the  radical  sign. 
We  shall  show  the  application  of  these  formulas,  and  those  in 
equations  {S),  hereafter. 

From  (30),  plane  trigonometry,  we  have 

sin.^=2  sin.^^  cos.-J^ 
Squaring,         .      sin.^u4=4  sin.^-J-^  cos.^-J^^  {t) 

Square  the  first  equation  in  {T),  and  multiply  it  by  the  square 
of  the  first  equation  in  (  U)y  and  four  times  their  product  is 

4  sin.»M  COS.' iA=    ^ "'"'"^  Bin.( ^-a)sin.( ^-i)sin.( ,S^) 

sin.^6  sin.^c 

Comparing  the  first  member  with  equation  (^),  we  have 


8in. 


sin.^6  sin.^c 


By  operating  in  the  same  manner  with  the  several  equations  in 
{T)  and  (  ?7),  we  have 

_     4  i?*sin.iS^sin.(AS' — a)sin.(^ — h)&m.(S — c)       ,  . 

£— V        n  .       i ^ (V) 

oi-r*    «   n    c•^■n    *  /»  »      ' 


Sin. 


sm.'  a  sin."  c 


The  numerators  of  the  second  members  of  (w)  and  (v),  are  the 
same ;  and  if  we  divide  (w)  by  (v),  and  extract  the  square  root, 

we  shall  have  sin.-4 sin.a 

sin.jB     sin.6 

Or,         .         .      sin.^  :  sin.u4=sin.J  :  sin.a,    a  truth  that  was 
demonstrated  in  proposition  9,  spherical  trigonometry. 


196  •  SPHERICAL 

We  have  again  demonstrated  it  in  this  manner,  to  show  that 
equation  (J^),  from  which  all  the  preceding  equations  arose,  is 
really  the  fundamental  equation  of  spherical  trigonometry. 

A  spherical  triangle  consists  of  six  parts  ;  three  sides,  and  three 
angles ;  and  there  are  certain  relations  existing  between  them  ;  but 
the  combinations  of  these  relations  have  their  limits  ;  and  when 
we  have  gone  through  these  relations,  if  we  continue  to  combine 
equations,  we  shall  only  fall  on  truths  previously  demonstrated, 
and  this  is  exemphfied  by  our  last  operations. 

APPLICATION. 

SOLUTION     OF    RIGHT     ANGLED     SPHERICAL 
TRIANGLES. 

1.  At  a  certain  time  the  sun's  longitude  was  40°  29'  30",  and  the 
obliquity  of  the  ecliptic  23°  27'  32".     What  was  the  declination  1 

Ans.  14°  68'  62". 

This  example  presents  a  right  angled  spherical  triangle,  A5C.    The 
hypotenuse,  AC=40°  29'  30",  and  the  angle 
A=23°  27'  32",  and  the  side,  CB,  is  required. 
By  our  system  of  notation,  AC=b,  BC=a. 

This  can  be  solved  by  equation  (3)  or  (13), 
which  are  essentially  the  same  ;  that  is. 

JR  8in.a=8in.2>  sin. A 
8in.&=sin.40°  29'  30"         .        9.812470 
8in.A=sin.230  27'  32"         .        9.599985 
Ans,  sin.a=sin.l4°  68'  62"         .         9.412455 
Rejecting  10  in  the  index,  is  the  same  as  dividing  by  the  radius,  as 
the  equation  requires. 

2.  At  a  certain  time,  the  difference  between  the  longitude  of  the  sun 
and  moon,  was  76°  10'  20",  and  the  moon's  latitude,  at  the  same  time, 
was  5°  9'  12"  north.  What  was  the  true  angular  distance  between 
the  centers  of  the  sun  and  moon  ?  Ans.  76°  13'  45". 

This  problem  presents  a  right  angled  spherical  triangle,  whose  base 
A5=76o  10'  20",  and  perpendicular  5C=5°  9'  12".  The  hypotenuse 
AC,  is  required.     Equation  (8)  or  (18)  solves  it. 

c=76°  10'  20"     COS.     .     9.378406 
a=  5°    9'  12"     COS.     .     9.998241 

6=76°  13'  45"     COS.     .     9.376647 


TRIGONOMETRY.  197 

3.  An  astronomer  observed  the  sun  to  pass  his  meridian  on  a  certain 
day  when  his  astronomical  clock  gave  2  h.  9  min.  33  sec.  for  the  sideriai 
time,  and  the  altitude  was  such  as  to  give  the  declination  of  13°  6'  6" 
north.  What  was  the  sun's  longitude,  and  what  was  the  obliquity  of 
the  ecliptic  ]  Ans.  Lon.  34°  39'  46".     Obliq.  eclip.  23°  27'  26". 

This  problem  presents  a  right  angled  spherical  triangle,  giving  its 
base  and  perpendicular,  and  demanding  the  hypotenuse,  and  the  angle 
at  the  base. 

2  h.  9  m.  33  s.=c=32°  23  16  cos.  .  9.926671 
a=13  6  6  COS.  .  9.988576 
i>=34    39  46        COS.     .     9.916146 

To  find  A,  we  apply  equation  (3)  or  (13),  as  they  are  one  and  the 

same. 

Rsm.a         .         .         .  19.364869 

sin.ft      (subtract)      .     9.754918 
.    A=23°27'26"    .         .     9.699951 

At  a  certain  time  the  sun's  longitude  will  be  160°  33'  20",  and  the 
obliquity  of  the  ecliptic  23°  27'  29".  Required  its  right  ascension  and 
declination.  Ans.  R.  A.  162°  37'  28";     Dec.  \\^  17'  7"N. 

Observation.  This  problem  presents  a 
right  angled  spherical  triangle,  whose  base 
and  hypotenuse  are  each  greater  than  90°  ; 
and  in  cases  of  this  kind,  let  the  pupil  ob- 
serve, that  the  base  is  greater  than  the  A^o- 
ie7tw.se,  and  the  oblique  angle  opposite  the  base,  is  greater  than  a  right 
angle.  In  all  cases,  a  triangle  and  its  supplemental  triangle,  make  a 
lun£.  It  is  180°  from  one  pole  to  its  opposite,  whatever  great  circle  be 
traversed.  It  is  180°  along  the  equator  AjBA',  and  also  180°  along  the 
ecliptic  ACA';  and  the  lune  always  gives  two  triangles;  and  when 
the  sides  of  one  of  them  are  greater  than  90°,  we  take  its  supplemental 
triangle,  as  in  this  case  we  operate  on  the  triangle  A'CB. 

But  A'C  is  greater  than  A'B;  therefore,  AB  is  greater  than  AC. 
The  angle  A'CB  is  less  than  90°;  therefore,  ACB  is  greater  than 
90°,  because  the  two  angles  together  make  two  right  angles. 

These  facts  are  technically  expressed,  by  saying,  that  the  sides  and 
opposite  angles  are  of  the  same  affection*;  and  if  the  two  sides  of  a 
right  angled  spherical  triangle  are  of  the  same  affection,  the  hypotenuse 

*  Same  affection :  that  is,  both  greater,  or  both  less  tlian  90*^.  Vffe  cut 
i^ection :  the  one  greater,  the  other  less  than  90°. 


196 


SPHERICAL 


will  be  less  than  90°;  and  of  different  affection,  the  hypotenuse  will  be 
greater  than  90°. 

If,  in  every  instance,  we  make  a  natural  construction  of  the  figure 
and  use  common  judgment,  it  will  be  impossible  to  doubt  whether  an 
arc  must  be  taken  greater  or  less  than  90°. 

We  now  solve  the  triangle  A'CB,    A'C=29°  26'  40". 


To  find  BC.    Eq.  (3)  or  (13).     h  sin.  29°  26'  40' 

A  sin.  23°  27'  29" 


a  sin.  11°  IT    T 
To  find  A'Bf  we  use  equation  (1)  or  (11),  thus  : 

ten.     11°  17'    7"     .       9.300016 
10.362674 


9.691594 
^.699984 
9.291578 


cot.     23°  27'  29' 
c  sin.     27°  22'  32' 
180 


9.662590 


A5=152°  37'  28' 


We  select  the  following  examples  to  exercise  the  pupils  in  right 
Angled  spherical  trigonometry: 

1.  In  the  right  angled  spherical  triangle 
AJ5C,  given  AB  118°  21'  4",  and  the  angle 
A  23°  40'  12",  to  find  the  other  parts. 

Am,  AC  116°  17'  55",  the  angle  C  100° 
69  26",  and  5C  21°  6' 42". 

2.  In  the  right  angled  spherical  triangle  ABC,  given  AB  53°  14' 
20",  and  the  angle  A  91°  25'  63",  to  find  the  other  parts. 

Ans.  AC  91°  4'  9",  the  angle  C  53°  15'  8",  and  J5C  91°  47'  11". 

3.  In  the  right  angled  spherical  triangle  ABC,  given  AB  102°  50' 
25",  and  the  angle  A  113°  14'  37",  to  find  the  other  parts. 

Ans.  AC  84°  51'  36",  the  angle  C  101°  46'  57",  and  BC  113° 
46'  27". 

4.  In  the  right  angled  shpherical  triangle  ABC,  given  AB  48°  24' 
16",  and  BC  59°  38'  27",  to  find  the  other  parts. 

Ans.  AC  70°  23'  42",  the  angle  A  66°  20'  40",  and  the  angle  C  52° 
32' 55". 

6.  In  the  right  angled  spherical  triangle  ABC,  given  AB  151°  23' 
9",  and  J5C  16°  35'  14"  to  find  the  other  parts. 

Ans.  AC  147°  16'  61",  the  angle  C  117°  37'  21",  and  the  angle  A 
31°  52'  60". 


TRIGONOMETRY.  199 

6.  In  the  right  angled  spherical  triangle  ABC,  given  AB  73°  4' 
31",  and  AC  86°  12'  15",  to  find  the  other  parts. 

Ans.  BC  76°  61'  20",  the  angle  A  77°  24'  23",  and  the  angle  C  73° 
29'  40". 

7.  In  the  right  angled  spherical  triangle  ABC,  given  AC  118°  32' 
12",  and  AB  47°  26'  35",  to  find  the  other  parts. 

Ans.  BC  134°  66'  20",  the  angle  A  126°  19'  2",  and  the  angle  C 
66°  68'  44". 

8.  In  the  right  angled  spherical  triangle  ABC,  given  AB  40°  18' 
23",  and  AC  100°  3'  7",  to  find  the  other  parts. 

Ans.  The  angle  A  98°  38'  53",  the  angle  C  41°  4'  6",  and  5C  103° 
13' 52". 

9.  In  the  right  angled  spherical  triangle  ABC,  given  A  C  61°  3'  22", 
and  the  angle  A  49°  28'  12",  to  find  the  other  parts. 

Ans.  AB  49°  36'  6",  the  angle  C  60°  29'  19",  and  BC  41°  41'  32". 

10  In  the  right  angled  spherical  triangle  ABC,  given  AB  29°  12' 
60",  and  the  angle  C  37°  26'  21",  to  find  the  other  parts  ] 

Ans.  Ambiguous ;  the  angle  A  65°  27'  58"  or  its  supplement,  A  C 
63°  24'  13"  or  its  supplement,  BC  46°  55'  2"  or  its  supplement. 

11.  In  the  right  angled  spherical  triangle  ABC,  given  AB  100°  lO' 
3",  and  the  angle  C  90°  14'  20",  to  find  the  other  parts. 

Ans.  Ambiguous  ;  AC  100°  9'  55"  or  its  supplement,  BC  1°  19'  53" 
of  its  supplement,  and  the  angle  A  1°  21'  8"  or  its  supplement. 

12.  In  the  right  angled  spherical  triangle  ABC,  given  AB  54°  21' 
35",  and  the  angle  C  61°  2'  15",  to  find  the  other  parts. 

Ans.  Ambiguous;  BC  129°  28'  28"  or  its  supplement,  AC  111° 
44'  34"  or  its  supplement,  and  the  angle  A  123°  47'  44"  or  its 
supplement. 

13.  In  the  right  angled  spherical  triangle  ABC,  gWen  AB  121° 
26'  25",  and  the  angle  C  111°  14'  37",  to  find  the  other  parts. 

Ans.  Ambiguous;  the  angle  A  136°  0'  3 'or  its  supplement,  A  C 
66°'  15'  38"  or  its  supplement,  and  BC  140°  30'  56"  or  its  supplement. 

The  solution  of  right  angled  spherical  tri- 
angles includes,  also,  the  solution  of  quad- 
rantal  triangles,  as  may  be  seen  by  inspecting 
the  adjoining  figure.  WTwn  we  have  one 
quadrantal  triangle,  we  have  four,  which  JiU  up 
the  whole  hemisphere. 

To  effect  the  solution  of  either  of  the  four 
quadrantal  triangles  APC,  AP'C,  A' PC,  or 


200  SPHERICAL 

A'P'  C,  it  is  sufficient  to  solve  the  small  right  angled  spherical  triangle 
ABC. 

To  the  half  lune  AP'B,  we  add  the  triangle  ABC,  and  we  have  the 
quadrantal  triangle  AP'  C;  and  by  subtracting  the  same  from  the  equal 
half  lune  APB,  we  have  the  quadrantal  triangle  PA  C. 

When  we  have  the  side,  A  C,  of  the  same  triangle,  we  have  its  sup- 
plement, A' C,  which  is  a  side  of  the  triangle  A' PC,  and  of  A'P'C. 
When  we  have  the  side,  CB,  of  the  small  triangle,  by  adding  it  to  90°, 
we  have  P'  C,  a  side  of  the  triangle  A'P'  C;  and  subtracting  it  from 
90°,  we  have  PC,  a  side  of  the  triangle  APC,  and  A' PC. 


EXAMPLES. 

\.  Ina  quadrantal  triangle,  there  are  given  the  quadrantal  side,  90°,  a  side 
adjacent,  42°  21',  and  the  angle  opposite  this  last  side,  equal  to  36°  31'. 
Required  the  other  parts. 

By  this  enumeration  we  cannot  decide  whether  the  triangle  APC  or 
AP'C,  is  the  one  required,  for  A  C=42°  21'  belongs  equally  to  both 
triangles.    The  angle  APC=AP'C=Z6°  Z1'=AB. 

We  operate  wholly  on  the  triangle  ABC. 

To  find  the  angle  A,  call  it  the  middle  part. 

Then,         R  cos.(  CAB)=R  sin.PA  C=cot.A  C.t&n.AB 

cot.AC=  42°21'  .   10.040231 

tan.A5=  36  31  .    9.869473 
cos.CAJ?=  35  40  61     9.909704 
90 


PAC=   64  19  9 
P'AC=125  40  61 


To  find  the  angle  C,  call  it  the  middle  part. 

R  COS.  A  CjB=sin.  CAB  cos. AS 

sm.CAB=   35°  40  51"    9.766869 

cos.AjB=  36  31    .    9.905085 

cos.ACB=   62   2  45     9.670954 

180 

A  CP=A'CP'=:1 17  67  15 


TRIGONOMETRY.  ml 

To  find  the  side  J?C,  call  it  the  middle  part, 

R  sm.BC=tSin.AB  cot. A CB, 

tan.AJ?=  36°  31'  0"  9.869473 

cot.ACB=  62      2' 45"  9.724835 

sm,BC=   23   8'  11"  9.694308 

90 

PC=   66  61'  49" 
P'C=113   8'  11" 

We  now  have  all  the  sides,  and  all  the  angles  of  the  four  triangles 
in  question. 

2.  In  a  quadrantal  spherical  triangle,  having  given  the  quadrantal  side, 
90°,  an  adjacent  side,  115°,  09',  and  the  included  angle,  116°  66',  to  find 
the  other  parts. 

This  enunciation  clearly  points  out  the 
particular  triangle  A' PC.  A'P'=90°;  and 
conceive  A'C=115°  09'.  Then  the  angle 
P'A'C=115°65'=P'i). 

From  the  angle  P'A'C  take  90°  or  P'A'B, 
and  the  remainder  is  the  angle  OA'D=BA  C 
=25°  55'. 

,We  here  again  operate  on  the  triangle 
ABC.    A'C  taken  from  180°, gives     .        .       64°6l'=AC 

To  find  BC,  we  call  it  the  middle  part. 

R  sin. jBC=sin. AC  sin.^AC. 

sin.AC=  64°  61'    .    9.956744 
sin.5AC=  25  55'    .    9.640544 

sin.jBC=  23  18'  19"     8.597288 
90 

P'C=113    18'  19" 
To  find  AB  we  call  it  the  middle  part. 

R  sin.AJB=tan.BC  cot.BAC. 


tan.J?C=  23°  18'  19" 
cot.5AC=  26  65'  . 
Bm.AB=   62  26'  8' 
180 


9.634251 
9.313423 

9.947674 


A'B=ll1    33'  62"=the  angle  A'P'C 


202  SPHERICAL 


To 

find 

the 

angle    C, 

we 

call 

it  the 

middle  part. 

R  cos.C= 

=cot.AC  tam.BC 

cot.  A  C= 

:  64°  61' 

. 

9.671634 

t3Ln.BC= 

:  23 

18' 

19"     . 

9.634251 

COS.  C= 

:    78 

180 

19' 

63" 

9.306885 

P'CA'= 

:101 

40' 

7" 

Thus  we  have  found  the  side  P'C=  11 3°  18'  19"  ) 

The  angle  A'P'C=  117°  33'  62"  }  Ans. 
"       P'CA'=10lo  40'    7"  > 

3.  In  a  quadrantal  triangle,  given  the  quadrantal  side,  90°,  a  side 
adjacent,  67°  3',  and  the  included  angle,  49°  18',  to  find  the  other 
parts. 

Ans.  The  remaining  side  is  63°  6'  46",  the  angle  opposite  the  quad- 
rantal side,  108°  32'  27",  and  the  remaining  angle,  60°  48'  54". 

4.  In  a  quadrantal  triangle,  given  the  quadrantal  side,  90°,  one  angle 
adjacent,  118°  40'  36",  and  the  side  opposite  this  last  mentioned  angle, 
113°  2'  28",  to  find  the  other  parts. 

Ans.  The  remaining  side  is  64°  38'  67",  the  angle  opposite,  61° 
2'  36",  and  the  angle  opposite  the  quadrantal  side  is  72°  26'  21". 

5.  In  a  quadrantal  triangle,  given  the  quadrantal  side,  90,  and  the 
two  adjacent  angles,  one  69°  13'  46",  the  other  72°  12'  4",  to  find  the 
other  parts. 

Ans.  One  of  the  remaining  sides  is  70°  8'  39",  the  other  is  73°  17' 
29",  and  the  angle  opposite  the  quadrantal  side  is  96°  13'  23". 

6.  In  a  quadrantal  triangle,  given  the  quadrantal  side,  90°,  one  adja- 
cent side,  86°  14'  40",  and  the  angle  opposite  to  that  side,  37°  12'  20", 
to  find  the  other  parts. 

Ans.  The  remaining  side  is  4°  43'  2",  the  angle  opposite,  2°  61'  23", 
and  the  angle  opposite  the  quadrantal  side,  142°  42'  2". 

7.  In  a  quadrantal  triangle,  given  the  quadrantal  side,  90°,  and  the 
other  two  sides,  one  118°  32'  16",  the  other  67°  48'  40",  to  find  the 
other  parts — the  three  angles. 

Ans.  The  angles  are  64°  32'  21",  121°  3'  40",  and  77°  11'  6";  the 
greater  angle  opposite  the  greater  side,  of  course. 

8.  In  a  quadrantal  triangle,  given  the  quadrantal  side,  90°,  the  angle 
opposite,  104°  41'  17",  and  one  adjacent  side,  73°  21'  6",  to  find  the 
other  parts. 

Ans.  The  remaining  side  is  49°  42'  18",  and  the  remaining  angles 
are  47°  32'  39",  and  67°  66'  13". 


TRIGONOMETRY.  203 

OBLiaUE    ANGLED     SPHERICAL 
TRIGONOMETRY. 

All  cases  of  oblique  angled  spherical  trigonometry  may  be  solved  by 
right  angled  trigonometry,  except  two  ;  because  every  oblique  angled 
spherical  triangle  is  composed  of  the  sum  or  difference  of  two  right 
angled  spherical  triangles. 

When  a  side  and  two  of  the  angles,  or  an  angle  and  two  of  the  sides 
are  given,  to  find  the  other  parts,  conform  to  the  following  directions  : 

Let  a  perpendicular  be  drawn  from  an  extremity  of  a  given  side,  and 
opposite  a  given  angle  or  its  supplement ;  this  will  form  two  right 
angled  spherical  triangles  ;  and  one  of  them  will  have  its  hypotenuse 
and  one  of  its  adjacent  angles  given,  from  which  all  its  other  parts  can 
be  computed  ;  and  some  of  these  parts  will  become  as  known  parts  to 
the  other  triangle,  from  which  all  its  parts  can  be  computed. 

To  facilitate  these  computations,  we  here  give  a  summary  of  the 
practical  truths  demonstrated  in  the  foregoing  propositions. 

1.  The  sines  of  the  sides  of  spherical  triangles  are  proportional  to  the 
sines  of  their  opposite  angles. 

2.  The  sines  of  the  segments  of  the  base,  made  by  a  perpendicular  from  the 
opposite  angle,  are  proportional  to  the  cotangents  of  their  adjacent  angles. 

3.  The  cosines  of  the  segments  of  the  base  are  proportional  to  the  cosines 
of  the  adjacent  sides  of  the  triangle. 

4.  The  tangents  of  the  segments  of  the  base  are  proportional  to  the 
the  tangents  of  the  opposite  segments  of  the  vertical  angles. 

6.  The  cosines  of  the  angles  at  the  base,  are  proportional  to  the  sines 
of  the  corresponding  segments  of  the  vertical  angles. 

6.  The  cosines  of  the  segments  of  the  vertical  angles  are  propoi  tional 
to  the  cotangents  of  the  adjoining  sides  of  the  triangle. 

The  two  cases  in  which  right  angled  triangles  are  not  used,  are, 
1st.  When  the  three  sides     are  given  to  find  the  angles  ;  and, 
2d.    When  the  three  angles  are  given  to  find  the  sides. 
The  first  of  these  cases  is  the  most  important  of  all,  and  for  that 

reason  great  attention  has  been  given  to  it,  and  two  series  of  equations, 

(T)  and  (I/),  have  been  deduced  to  facilitate  its  solution. 
We  now  apply  the  following  equation  to  find  the  angle  A,  of  the 

triangle  AJB  C,  whose  sides  are  a,  b,  c.    a==70°  4'  18".    b=6Z°  21'  27". 

00:590  iQ'  23".    a  is  opposite  A,  b  is  opposite  B,  and  c  is  opposite  C« 


204  *  SPHERICAL 


IR^  sin.S  sin.(iS— a) 
^        N         sm.o  sm.c 
We  write  the  second  member  of  this  equation  thus : 

showing  four  distinct  logarithms. 

R 

The  logarithm  corresponding  to  -; — r    is  the  sin.ft  subtracted  from 

R 

10 :  and    -: —    is  the    sin.c  subtracted  from    10,  which    we    call 
sin.c 

8tn.complement. 

BC=a=  70°    4'  18" 

AB==c=  59°  16'  23"  sin.com.     0.065697' 

AC=b=  63°  21'  27"  sin.com.      0.048749 

2)192    42     8 

5:=  96    21     4"  sin.  9.997326 

8—a=  26    16     46  sin.  9.646158 

2)19.757930 

iA=  40    49   10  cos.    9.878965 
2_ 

A=  81    38  20 
When  we  apply  the  equation  to  find  the  angle  A,  we  write  a  first,  at 
the  top  of  the  column  ;  when  we  apply  the  equation  to  find  the  angle 
B,  we  write  b  at  the  top  of  the  column.    Thus, 
To  find  tlie  angle  B 


fl22  sm.S  8m.i&—b) 
sin.fl  sm.c 


.i5=V- 


h=  63°  21'  27" 

c=  59    16  23    sin.com.  .     .065697 

a=  70      4   18     sin.com.  .     .026857 

2)192    42     8 

S=  96    21     4    sin.    .  .  9.997326 

S—l=  32    59  37     sin.    .  .  9.736034 

2)19.825874 

iB=  35      4  49    cos.    .  9.912937 
2 


B=  70      9  38 


TRIGONOMETRY.  205 

By  the  other  equation  in  formula  (T),  we  can  find  the  angle  C;  but, 
for  the  sake  of  variety,  we  will  find  the  angle  C  by  the  application  of 
the  third  equation  in  formula  (t/). 


/jR2  sin.(S— 6)  sin.(S— a) 
sin.  '^  — — 


Mt-^V     8in.i6in.a 

=^/(sL)(sL)^-(*-^)  «-(*-: 

c—   590  16'  23" 

, 

0=  70   4  18  sin.com. 

.026817 

6=  63  21  27  sin.com. 

.048479 

2)192  42  8 

8=   96  21  4 

8— a—  26  16  46  sin. 

.   9.646168 

/S— 5=  32  69  37  sin. 

.   9.736034 

2)19.467768 

JC=  32°  23'  17"  sin. 
2 

.   9.778879 

C=  64    46  34 

To  show  the  harmony  and  practical  utility  of  these  two  sets  of 
equations,  we  will  find  the  angle  A,  from  the  equation 


sin 


-iW(sl)(i.c)^-(^>^-(^ 

a=  70   4'  18" 

6=  63  21  27  sin.com. 

c—   69  16  23  sin.com. 

.048749 
.066697 

2)192  42  8 

«=  96  21  4 
8—b=   32  69  37  sin.  . 
iS— c=  37   4  41  sin.  . 

.  9.736034 

.  9.780247 
2)19.630727 

iA=  40°  49'  10"  sin.  . 
2 

.  9.816363 

A=  81    38  20 

2.  In  a  spherical  triangle  ABC,  given  the  angle  A,  38°  19'  18",  the 
angle  J5, 48°  0'  10",  and  the  angle  C,  121°  8'  6",  to  find  the  sides  a,  b,  c. 
Apply  proposition     5,    spherics. 


206  SPHERICAL 

i4=  38°  19'  18"  supplement  141°  40'  42" 
B=  48  0  10  supplement  131  69  50 
C=121      8     6      supplement     58    51  64 

We  now  find  the  angles  to  the  spherical  triangle,  whose  sides  are 
these  supplements. 

Thus, 


UP  40'  42" 

131  59  50 

sin.com. 

*   .128909 

58  61  54 

sin.com. 

.067651 

2)332  32  26 

166  16  13 

sin. 

9.375376 

24  36  31 

sin. 

9.619253 
2)19.191088 

66°  47'  37i' 

'  cos.  . 

9.695543 

2 

angle  =133    35   15 

supp.  =  46    24  45=a  of  the  original  triangle. 
In  the  same  manner  we  find  6=60°  14'  25"    c=89°  1'  14" 

EXAMPLES      EOK      EXEECISE. 

1.  In  any  triangle,  ABC,  whose  sides  are  a,  b,  c,  given  6=118^' 
14",  c=120°  18'  33",  and  the  included  angle  A=27°  22'  34",  to  find 
tlie  other  parts. 

A71S.  a=23°  67'  13",  angle  B—dl^  26'  44",  and  C=102°  6'  54". 

2.  Given  A=81°  38'  17",  5=70°  9'  38",  and  C=64°  46'  32",  to  find 
the  sides  a,  b,  and  c. 

Ans,  a=70°  4'  18",  6=63°  21'  27",  and  c=59°  16'  23". 

3.  Given  the  three  sides  a=93°  27'  34",  6=100°  4'  26",  and  c=96° 
14'  50",  to  find  the  angles  A,  B,  and  C. 

Ans.  A=94°  39'  4",  5=100°  32'  19",  and  C=96°  68'  36". 

4.  Given  two  sides,  6=84°  16',  c=81°  12',  and  the  angle  C=80° 
28',  to  find  the  other  parts. 

Ans.  The  result  is  ambiguous,  for  we  may  consider  the  angle  B  as 
acute  or  obtuse.  If  the  angle  B  is  acute,  then  A=97°  13'  45", 
5=83°  11'  24",  and  a=96°  13'  33". 

If  B  is  obtuse,  then  A=21°  16'  44",  5=96°  48'  36",  and 
a=21°  19'  29" 

*  The  sine  complement  of  131^  59'  50",  is  the  same  as  the  sine  complement 
of  48°  0' 10". 


TRIGONOMETRY.  207 

6.  Given  one  side,  cz=64°  26',  and  the  angles  adjacent,  A=49°,  and 
5=52°,  to  find  the  other  parts. 

Ans.  6=45°  56'  46",  a=43°  29'  49",  and  C=98°  28'  5". 
6    Given  the  three  sides,  a=90°,  6=90°,  c=90°,  to  find  the  angles 
A,  B,  and  C.  Ans,  A=90°,  J5=90°,  and   C=90°. 

7.  Given  the  two  sides,  a=77°  25'  11",  and  c=128°  13'  47",  and 
the  angle  C,  131°  11'  12"  to  find  the  other  parts. 

Ans.  6=84°  29'  24",  A=69°  14',  and  5=72°  28'  46'. 

8.  Given  the  three  sides,  a,  6,  c,  a=68°  34'  13",  6=59°  21'  18, 
and  c=112°  16'  32",  to  find  the  angles  A,  J3,  and   C. 

Ans.  A=45°  26'  12",  5=41°     11'     6",  C=134°  54'  27" 

APPLICATION. 

Spherical  trigononometry  becomes  a  science  of  incalculable 
importance  in  its  connection  with  geography,  navigation,  and 
astronomy;  for  neither  of  these  subjects  can  be  understood  without 
it ;  and  to  stimulate  the  student  to  a  study  of  the  science,  we  here 
attempt  to  give  him  a  glimpse  at  some  of  its  points  of  application. 

Let  the  lines  in  the  an- 
nexed figure  represent  cir- 
cles in  the  heavens  above 
and'  around  us. 

Let  Z  be  the  zenith,  or 
the  point  just  overhead,  Hch 
the  horizon,  PZII  the  meri- 
dian in  the  heavens,  P  the 
pole  of  the  earth's  equator ; 
then  Ph  is  the  latitude  of 
the  observer,  and  PZ  is  the 
co.latitude.  Qcq  is  a  portion 
of  the  equator,  and  the  dotted,  curved  line,  m!S' S,  parallel  to  the 
equator,  is  the  parallel  of  the  sun's  declination  at  some  particular 
time ;  and  in  this  figure  the  sun's  declination  is  supposed  to  be 
north.  By  the  revolution  of  the  earth  on  its  axis,  the  sun  is 
apparently  brought  from  the  horizon,  at  Sy  to  the  meridian,  at  m  : 
and  from  thence  it  is  carried  down  on  the  same  curve,  on  the  other 
side  of  the  meridian  ;  and  this  apparent  motion  of  the  sun  (or  any 
other  celestial  body)  makes  angles  at  the  pole  P,  which  are  in 
direct  proportion  to  their  times  of  description. 


208  SPHERICAL 

The  apparent  straight  line,  Zc,  is  what  is  denominated,  in  astro- 
nomy, ^e  prime  vertical;  that  is,  the  east  and  west  line  through  the 
zenith,  passing  through  the  east  and  west  points  in  the  horizon. 

When  the  latitude  of  the  place  is  north,  and  the  declination  is 
also  north,  as  is  represented  in  this  figure,  the  sun  rises  and  seta 
on  the  horizon  to  the  north  of  the  east  and  west  points,  and  the 
distance  is  measured  by  the  arc  cS,  on  the  horizon. 

This  arc  can  be  found  by  means  of  the  right  angled  spherical 
triangle  cqSy  right  angled  at  q.  Sq  is  the  sun's  declination,  and  the 
angle  Scq  is  equal  to  the  codatitvde  of  the  place  ;  for  the  angle  Pch 
is  the  latitude,  and  the  angle  Scq  is  its  complement. 

The  side  cq^  a  portion  of  the  equator,  measures  the  angle  cPq^ 
the  time  of  the  sun's  rising  or  setting  before  or  after  six,  apparent 
time.  Thus  we  perceive  that  this  little  triangle  cSq,  is  a  very 
important  one. 

When  the  sun  is  exactly  east  or  west,  it  can  be  determined  by  the 
triangle  ZPS';  the  side  PZ  is  known,  being  the  co.latitude;  the  angle 
PZS'  is  a  right  angle,  and  the  side  PS'  is  the  sun's  polar  distance. 
Here,  then,  is  the  hypotenuse  and  side  of  a  right  angled  spherical 
triangle  given,  from  which  the  other  parts  can  be  computed.  The 
angle  ZPS'  is  the  time  from  noon,  and  the  side  ZS'  is  the  sun's 
zenith  distance  at  that  time. 

FORMULA      FOR      TIME. 

The  most  important  problem  in  navigation  is  that  of  finding  the 
time  from  the  altitude  of  the  sun,  when  the  sun's  declination  and 
the  latitude  of  the  observer  are  given. 

This  problem  will  be  un- 
derstood by  the  triangle 
PZS.  When  the  sun  is  on 
the  meridian,  it  is  then  ap- 
parent noon.  When  not  on 
the  meridian,  we  can  de- 
termine the  interval  from 
noon  by  means  of  the  tri- 
angle PZS;  for  we  can 
know  all  its  sides ;  and  the 
angle  at  Py  changed  into 
time  at  the  rate  of  15°  to 


TRIGONOMETRY.  209 

one  hour,  will  give  the  time  from  apparent  noon,  when  any  par- 
ticular altitude,  as  TS^  may  have  been  observed.  FS  is  known 
by  the  sun's  declination  at  about  the  time  ;  and  PZ  is  known,  if 
the  observer  knows  his  latitude. 

Having  these  three  sides,  we  can  always  find  the  sought  angle 
at  the  pole,  by  the  equations  already  given  in  formulas  (T),  or 
{U)\  but  these  formulas  require  the  use  of  the  co.latitude  and  the 
CO. altitude,  and  the  practical  navigator  is  very  averse  to  taking  the 
trouble  of  finding  the  complements  of  arcs,  when  he  is  quite 
certain  that  formulas  can  be  made,  which  comprise  but  the  arcs 
themselves. 

The  practical  man,  also,  very  properly  demands  the  most  concise 
practical  results.  No  matter  how  much  labor  is  spent  in  theoriz- 
ing, provided  we  arrive  at  practical  brevity  ;  and  for  the  especial 
accommodation  of  seamen,  the  following  formula  for  finding  time 
has  been  deduced. 

From  the  fundamental  equation  of  spherical  trigonometry,  taken 
from  page  191  we  have, 

cos.ZS—cos.PZ  coa.PS 


cos.P=- 


sin.PZ  sin.P/S' 


Now,  in  place  of  cos. ZS,  we  take  sm. ST,  which  is,  in  fact,  the 
same  thing,  and  in  place  of  cos. PZ,  we  take  sin.lat.,  which  is  also 
the  same. 

In  short,  let  A=:  the  altitude  of  the  sun,  L=  the  latitude  of  the 
observer,  and  D=  the  sun's  polar  distance. 

r^,  _.     sin.^ — sin.Z  cos.D 

ihen,  .         .     cos.P= ~ — ; — — 

cos./y  sm. JJ 

But,      .       2  sin.2  iP=\ — cos.P         (See  eq.  32,  page  143.) 

sin.^ — sin.Z  cos.  D 


Therefore,  2  sin.^  iP=l 


cos.L  siu.D 


(cos.Z  sin.Z>-}-sin.Z  cos.2))— -sin.-4 

cos.Z  sin.D 


_sin.  (L-]rD) — sin.^ 

cos.Z  sin.i> 
14 


210  SPHERICAL 

Considering  (L-^-D)   as  a  single   arc,  and  applying  equation 
(16),  plane  trigonometry,  we  have,  after  dividing  by  2, 

sm.2  iF=z r-^-n 

COS.//  sm.lJ 

L+D-A    L+D+A  '    ,  ., 

J)ut,     • = A    and  if  we  assume 


^_L-\-D-\-A,  we  shall  have. 


2 


.    „,„     cos-zS^sinY/S' — A) 

sm.«  iP= ^~\—  — -^ 

■*  cos.//  sm.x> 


^        -ID       hos.Ssm(S — A) 
Or,    sm.  iP=\ -~i — =—^ 

^  N  />ns   /^  sm    7) 


cos.Z  sin.i> 

This  is  the  final  result,  when  the  radius  is  unity,  and  when  the 
radius  is  greater  by  B,  then  the  sin.  -JP,  will  be  greater  by 
M;  and,  therefore,  the  value  of  this  sine,  corresponding  to  our 
tables  is. 


sm 


•*^=>/(3l7-)(s-i^)«°^-'^^'"(^-^) 


This  equation  is  known  as  the  sailor's  formula  for  time,  and  a 
very  concise  and  beautiful  formula  it  is  ;  it  is  used  by  thousands 
who  have  little  knowledge  of  how  it  is  obtained,  or  who  know 
little  of  the  amount  of  science  there  is  wrapt  up  in  it. 

When  the  observer  has  logarithmic  tables  that  contain  secants 
and  cosecants,  the  above  equation  can  be  modified. 

Because,       sec.Z= =:  andcosec.i>=-^ — ^ 

cos.//  sm.X' 

(See  equations,  plane  trigonometry,  page  138.) 


Therefore,  sin.|P=^(  '-^  ]  (  -^^  )  cos. ^ sin. (^S'— J) 

Here,  then,  we  have /o«r  distinct  logarithms  to  be  added  together 
and  divided  by  2,  which  is  extracting^^quare  root. 


TRIGONOMETRY.  211 

The  first  logarithm  is  the  secant  of  the  latitude,  diminished  by 
the  index  10 ;  the  second  is  the  cosecant  of  the  polar  distance, 
diminished  by  the  index  10  ;  the  third  is  the  cosine  of  the  half 
sum  of  altitude,  latitude,  and  polar  distance  ;  and  the  fourth  is  the 
sine  of  an  arc,  found  by  diminishing  this  half  sum  by  the  altitude. 
Navigators  retain  this  formula  in  memory  by  the  following  words : 
Altitude — latitude — polar  distance — half  sum — remainder;  secant 
— cosecant — cosine — sine. 

EXAMPLE. 

In  latitude  39°  6'  20"  north,  when  the  sun's  declination  was 
12°  3'  10",  north,  the  true  altitude*  of  the  sun's  center  was 
observed  to  be  30°  10'  40",  rising.    What  was  the  apparent  time  ? 

Alt.    30°   10' 30"       . 


Lat.     39 

6   20 

cos.com.     .110146 

P.D.    11 

bQ  60 

sin.c 

!om.    .009680 

2)147 

13  40 

S=     73 

36   60 

cos. 

.     9.460416 

(S—A)^     43 

26  20 

sin. 

.      9.837299 
2)19.407541 

30 

22     6 
2 

sin. 

9.703770 

P=      60    44   10 

This  angle,  converted  into  time,  at  the  rate  of  15°  to  one  hour, 
or  4  minutes  to  1°,  gives  4h.  2m.  56s.  from  apparent  noon  ;  and 
as  the  sun  was  rising,  it  was  before  noon,  or 

7h.  67m.  4s.     A.  M 

If  to  this  the  equation  of  time  were  given  and  applied,  we  should 
have  the  mean  time  ;  and  if  such  time  were  compared  to  a  cluck 
or  watch,  we  could  determine  its  error.  A  good  observer,  with  a 
good  instrument,  can,  in  this  manner,  determine  the  local  time 
within  4  or  6  seconds. 

*  The  instrument  used,  the  manner  of  taking  the  altitude,  its  correction 
for  refraction,  seraidiameter,  and  other  practical  or  circumstantial  details,  do 
not  belong  to  a  work  of  this  kind,  but  to  a  work  on  practical  astronomy  of 
navigation. 


212  SPHERICAL 

The  great  importance  of  determining  the  exact  time,  at  sea,  is 
to  determine  the  longitude,  which  is  but  the  difference  of  the  local 
time  between  the  observer's  meridian  and  the  assumed  prime 
meridian. 

A  timepiece,  of  nice  and  delicate  construction,  called  a  chrono- 
meter, by  its  rate  of  motion  and  adjustment,  will  show  the  time  at 
Greenwich,  or  at  any  other  known  meridian  to  which  it  refers  ;  and 
this  time,  compared  with  an  observation  on  the  sun,  will  determine 
the  amount  of  difiference  in  local  times,  which  is,  in  substance, 
longitude. 

The  same  triangle,  FZS,  gives  the  bearing  of  the  sun,  which  is 
is  called  its  azimuth ;  that  is,  the  angle  FZS  is  the  azimuth 
from  the  north,  and  its  supplement,  RZS,  is  its  azimuth  from  the 
south.  This  is  the  true  bearing ;  and  if  the  bearing  per  compass 
is  the  same,  then  the  compass  has  no  variation ;  if  different,  the 
amount  of  difference  gives  the  amount  of  the  variation  of  the 
compass. 

HOW    TO    MANAGE    A    LOCAL     SOLAR     ECLIPSE. 

We  shall  touch  this  subject  only  so  far  as  to  show  the  applica- 
tion and  utility  of  spherical  trigonometry. 

The  angular  semidiameter  of  the  sun  is  about  1 5',  and  that  of 
the  moon,  about  the  same  ;  and,  of  course,  when  an  eclipse  of  the 
sun  commences  or  ends,  the  apparent  distance  between  the  sun  and 
moon  cannot  be  greater  than  about  32',  or  a  little  more  than  half 
a  degree. 

The  nautical  almanac,  or  the  astronomical  tables,  will  give  us 
the  time  when  the  sun  and  moon  fall  into  line  on  the  same  meridian 
of  riff  hi  ascension,  and  give  us,  also,  their  dift'erence  in  declinations, 
at  the  same  time,  together  with  all  the  other  necessary  elements, 
such  as  semidiameters,  horizontal  parallax,  hourly  motions,  (fee. 

Now  let  us  take  the  time  when  the  moon  is  in  conjunction  with 
the  sun  in  right  ascension,  and  demand  the  apparent  distance 
between  the  centers  of  the  sun  and  moon,  as  seen  from  any 
particular  locality. 

By  the  time  as  given  in  the  nautical  almanac,  we  know  the 
sun's  distance  from  the  local  meridian,  either  east  or  west. 


TRIGONOMETRY.  21& 

Look  at  the  last  figure.  Let  S  represent  the  position  of  the 
Eun*s  center,  F  the  pole,  and  Z  the  zenith  of  the  observer. 

Then,  in  the  triangle  ZFS,  we  know  the  two  sides,  ZF  and  FS; 
and  from  the  apparent  time,  we  know  their  included  angle,  ZFS. 

The  declination  of  both  sun  and  moon  is  also  given  in  the  nau- 
tical almanac,  corresponding  to  this  time ;  and  their  diflference 
gives  the  space  which  we  represent  by  Sm,  on  our  figure.  From 
the  triangle  FZm  (two  sides  and  angle  inclnded),  compute  Zm  and 
the  angle  ZmF, 

The  efiFect  of  parallax  is  to  depress  the  body  in  a  vertical  direc- 
tion ;  and  if  m  is  its  true  place,  as  seen  from  the  center  of  the 
earth,  n  may  represent  its  apparent  place,  as  seen  by  the  observer, 
whose  zenith  is  Z. 

The  arc  mn  is  computed  from  the  horizontal  parallax,  by  the 
following  proportion,  p  representing  the  lunar  horizontal  parallax. 

Had.  :  cos.  3)  app.altitude  =p  :  mn. 

The  angle  Smn=ZmFy  and  the  angle  ZmF  is  computed  from 
the  triangle  FZm.  Now,  the  triangle  Smn  is  always  very  small ; 
the  sides  are  never  more  than  a  degree  in  length,  and  are  generally 
much  less ;  and  it  therefore  may  be  regarded  as  a  plane  triangle, 
with  two  sides,  Sm  and  mn,  and  the  angle  Smn,  between  them, 
given.  From  these  data  we  can  compute  the  distance  between  S 
and  n;  and  if  that  distance  is  less  than  the  sum  of  the  semidiame- 
ters  of  the  sun  and  moon,  the  sun  must  then  be  in  an  eclipse — 
otherwise  it  is  not. 

But  whether  the  distance  between  ;S^  and  n  is  less,  equal,  or 
greater  than  the  semidiameters  of  the  sun  and  moon,  by  it  the 
computer  can  assume  an  approximate  time  for  the  beginning  or  end 
of  the  eclipse,  as  the  case  may  be. 

In  case  the  computer  wishes  to  compute  the  apparent  distance 
between  sun  and  m<xm,  corresponding  to  any  other  time  than  that 
of  conjunction  in  ri^ht  ascension,  he  may  assume  any  interval  before 
or  after  that  period;  and  by  the  moon's  motion  from  the  sun  during 
that  interval,  he  can  put  the  moon  in  its  true  place,  at  m. 

Now,  by  the  help  of  the  spherical  triangle  FZm,  and  the  moon's 
horizontal  parallax,  the  distance  mn  can  be  computed  as  before ; 


214  SPHERICAL 

and  by  means  of  the  little  triangle  mna,  we  compute  the  distances 
na  and  am.  The  distance  na  is  parallax  in  right  ascension,  and  ma 
is  parallax  in  dechnation.  Parallax  increases  the  moon's  rio-ht 
ascension  when  the  moon  is  east  of  the  meridian,  and  diminishes 
it  when  west  of  the  meridian. 

Now,  the  difference  be- 
tween PS  and  Fa,  is  the 
apparent  difference  of  dec- 
lination of  the  sun  and  moon; 
and  nc  is  the  apparent  dif- 
ference of  right  ascension 
of  the  same  bodies ;  ca  is 
the  real  difference  in  right 
ascension.  The  distances 
Sc  and  en,*  expressed  in 
seconds  of  arc  as  linear  units, 
form  two  sides  of  a  right 
angled  plane  triangle ;  and 
the  distance  Sn,  the  hypotenuse,  is  the  apparent  distance  between 
the  center  of  the  sun  and  the  center  of  the  moon ;  and  just  at 
the  commencement  or  end  of  an  eclipse,  that  distance  will  be  equal 
to  the  semidiameter  of  the  sim,  added  to  the  semidiameter  of  the 
moon. 

But  it  would  be  only  accident  if  an  operator  should  assume  the 
exact  time  of  the  beginning  or  end  of  an  eclipse  ;  but  the  distance 
Sn,  computed,  would  indicate  whether  the  eclipse  had  already  com- 
menced or  ended,  or  would  commence  or  end  within  some  very 
short  interval  of  time. 

Astronomers,  however,  are  in  the  habit  of  taking  two  intervals 
of  time,  about  10  or  15  minutes  asunder,  between  which  they  know 
the  eclipse  will  commence,  and  compute  the  apparent  distance,  Suy 
for  these  two  periods  ;  one  of  them  will  be  less,  and  the  other 
greater  than  the  sum  of  the  two  semidiameters  ;  and  thus  they  find 
data  to  proportion  to  the  commencement  or  end  in  question. 

By  the  same  principles  astronomers  compute  the  beginning  and 
end  of  occultations. 

*  The  number  of  seconds  in  en  must  be  multiplied  by  the  cosine  of  the 
declination,  because  en  is  an  arc  of  a  small  circle. 


TRIGONOMETRY.  215 

MISCELLANEOUS  ASTRONOMICAL  EXAMPLES. 

1.  In  latitude  40°  48'  north,  the  sun  bore  south  78°  16'  west,  at 
Sli.  37m.  59s.  P.  M.,  apparent  time.  Required  his  altitude  and 
declination. 

Ans.  The  altitude  36°  46',  and  declination  15°  32'  north. 

2.  In  north  latitude,  when  the  sun's  declination  was  14°  20' 
north,  his  altitudes,  at  two  different  times  on  the  same  forenoon, 
were  43°  7'-!-,*  and  67°  lO'-j-;  and  the  change  of  his  azimuth,  in 
the  interval,  45°  2  .     Required  the  latitude.     Ans.  34°  20'  north. 

3.  In  latitude  16°  4'  north,  when  the  sun's  declination  is  23°  2' 
north.  Required  the  time  in  the  afternoon,  and  the  sun's  altitude 
and  bearing  when  his  azimuth  neither  increases  nor  decreases. 

Ans.  Time  3h.  9m.  26s.  P.  M.,  altitude  45°  1',  and  bearing 
north  73°  16'  west. 

4.  The  sun  set  south  west  ^  south,  when  his  declination  was 
16°  4'  south.     Required  the  latitude.  Ans.  69°  1'  north. 

5.  The  altitude  of  the  sun,  when  on  the  equator,  was  1 4°  2S'-\-, 
bearing  east  22°  30'  south.     Required  the  latitude  and  time. 

Ans.  Latitude  56"*  1',  and  time  7h.  46m.  12s.  A.  M. 

'  6.  The  altitude  of  the  sun  was  20°  41'  at  2h.  20m.  P.  M,  when 
his  declination  was  10°  28'  south.  Required  his  azimuth  and  the 
latitude.     Ans.  Azimuth  south  37°  5'  west,  latitude  51°  58'  north. 

7.  If,  on  August  11,  1840,  Spica  set  2h.  26m.  14s.  before  Arc- 
turus,  hight  of  the  eye  15  feet,  required  the  north  latitude. 

Ans.  38°  46'  north. 

8.  If,  on  November  14,  1829,  Menkar  rise  48m.  3s.  before 
Aldebaran,  hight  of  the  eye  17  feet,  required  the  north  latitude. 

Ans.  39°  33'  north. 

9.  In  latitude  16°  40'  north,  when  the  sun's  declination  was 
23°  1 8'  north,  I  observed  him  twice,  in  the  same  forenoon,  bearing 
north  68°  30'  east.  Required  the  times  of  observation,  and  his 
altitude  at  each  time. 

Ans.  Times  6h.  15ni.  40s.  A.  M.,  and  lOh.  32m.  48s.  A.  M., 
altitudes  9°  59'  36",  and  68°  29'  42". 

*  Plus  means  rising  ;  and,  of  course,  forenoon. 


216  •  SPHERICAL 


LUNAR    OBSERVATIONS. 

The  moon  revolves  througli  a  great  circle  of  the  celestial  sphere 
in  about  27  days  and  8  hours  ;  and  astronomers  are  able  to  desig- 
nate its  exact  position  in  respect  to  the  stars,  corresponding  to  any 
definite  time. 

But  the  observer  is  supposed  to  be  at  the 
center  of  the  earth.  The  moon  is  never  seen 
by  an  observer  in  exactly  its  true  planej  unless 
the  observer  is  in  a  line  between  the  center  of 
the  earth  and  the  center  of  the  moon ;  that  is, 
unless  the  moon  is  in  the  zenith  of  the  observer; 
in  all  other  positions  the  moon  is  depressed  by 
parallax,  and  appears  nearer  to  those  stars  which  are  below  her, 
and  further  from  those  that  are  above  her,  than  would  appear 
from  the  center  of  the  earth. 

The  true  distance  between  the  sun  and  moon,  or  between  a  star 
and  the  moon,  can  be  deduced  from  the  apparent  distance,  by  the 
application  of  spherical  trigonometry. 

The  apparent  altitudes  of  the  two  objects  must  be  taken,  and 
corrected  for  parallax  and  refraction. 

Let  Z  be  the  zenith  of  the  observer,  S'  the  apparent  place  of  the 
sun  or  star,  and  S  its  true  place  ;  also,  let  m'  be  the  apparent  place 
of  the  moon,  and  m  its  true  place,  as  seen  from  the  center  of  the 
earth. 

With  the  observed  sides  of  the  spherical  triangle  ZS'm',  we 
compute  the  angle  at  Z;  then, in  the  triangle  ZSmwa  have  the  two 
sides  ZS  and  Zm,  and  the  included  angle  at  Z,  from  which  we 
compute  the  side  Sm,  which  is  the  true  distance. 

To  the  definite,  true  distance,  there  is  a  corresponding  definite 
Gh-eenwick  time,  which  the  practical  navigator  can  find  with  the 
utmost  facility.  This  time  at  thej^rs^  meridian,  compared  with  the 
local  time  deduced  from  the  altitude  of  the  sun,  will  of  course  give 
the  longitude. 

To  deduce  the  true  distance  from  the  apparent,  is  called  working 
a  lunar,  and  is  a  subject  of  considerable  perplexity  to  the  young 
navigator;  but,  bj  means  of  auxiliary  tables,  and  rules  for  delicate 


TRIGONOMETRY.  217 

approximations,  science  and  art  have  nearly  overcome  all  difficul- 
ties, and  a  good  operator  can  now  work  a  lunar  in  about  five 
minutes. 

We  here  only  give  a  view  of  the  scientific  principles  involved. 
For  complete  practical  knowledge  we  must  consult  books  on 
navigation. 

APPENDIX     TO    TRIGONOMETRY. 

For  the  benefit  of  those  who  may  desire  to  cultivate  a  taste  for 
mathematical  science,  we  give  the  following  exercises,  which  are 
designed  to  strengthen  the  powers  for  geometrical  investigations. 

To  demonstrate  equations  (7),  (8),  (9), 
and  (10),  geometrical]^,  the  pupil  must  be 
fully  impressed  with  the  following  principles: 

1 .  An  angle  in  a  semicircle  is  a  right  angle, 

2.  If  one  side  of  a  right  angled  triangle  is 
made  the  sine  of  its  opposite  angle,  the  other 
side  tvill  be  the  cosine  of  the  same  angle* 

(See  proposition  3,  page  147.) 

3.  Any  chord  is  double  the  sine  of  half  the  arc.  (See  observation 
3.  page  138.) 

4.  Observe  theorem  21,  book  3. 

Now  from  A,  any  point  on  a  circle,  take  AB,  the  double  of  any 
arc  designated  by  a,  and  AC,  double  of  any  arc  designated  by  b. 
Draw  AD,  the  diameter,  and  consider  its  value  equal  2,  twice 
the  radius  of  unity.     Join  BD  and  DC. 

Tlien,  by  reason  of  the  quadrilateral  in  a  circle,  we  have, 
AD'BC=AB'DG+AC'BD  (1) 

But,         ,         ^^=2sin.a)      .,         AC=2  sin.b  i 
BD=2cos.a\        ^^'    DC=2  cos.b\ 
BC=2  sin.(a+6),  and  AD=2 
Substituting  these  values  in  ( 1 ),  we  have 

4  sin.(a-\-b)=2  sin.a  2  cos.b-\-2  cos.a  2  sin.5 
Dividing  by  4,  and 

8in.(a+J)=sin  a  cos.5-|-cos.a  sin.6 


218  APPENDIX   TO 

Now  let  the  arc  CAB=2a,  and  AB=2b;  then  AC=2a—2d 
And,         .      CB=2sm.a,       AC=2sm.(a — b),      £D=2cos.6 

AB^2  sm.b,       DC=:2  cos.(a--b} 
Substituting  these  values  in  equation  (1),  we  hare 

4sin.a=2  sin.5  2  cos. (a — i)+2  sin.(a — b)2  cos4 
Dividing  by  4,    sin.a=sin.i  cos.(a — 6)+sin.(a — b)cos,b 

To  demonstrate  equation  (8.)     Let  the 
&rc  AB=2a,  AC=2  b; 


Then,       .         BC=2(a—b) 

And,  by  reason  of  the  quadrilateral, 
AB'DC=BC'AD-\-AC*BD    (2) 


But, 


AB=2  sin. a 
BD 


=2  sin.a  )   ^j^^     A  (7=2  sm.b ) 
=2  008.01  '    i>C=2cos.4 


AI>=2,  and  BC=2  sin.(a— 5) 

These  values  substituted  above,  and  we  have 

2  sin.a  2  cos.5=4  sin.(a — b)-]-2  sin.b  2  cos.o 

Dividing  by  4,  transposing,  &c.. 

And       sin.(a — 5)=sin.a  cos.5 — sin.6  cos.a 

Again,  let  the  arc  AC=2a,  the  arc  CB—2b;  then  the  arc 

ACB=2(a-\-b), 
And  the  chord      AB=2  s]n.{aA-b)  )    A  C'=2  sin  a  i 
BD=2  cos.{a-\-b)  ]  J)C=2  cos.a  ) 

AD=2,  and  ^(7=2  sin.J 
Substituting  these  values  in  equation  (2),  we  have, 
2  cos.a  2  sin.(a-|-6)=4sin.6-|-2  sin.a  2  cos.(a-fi) 

Dividing  by  4, 

cos.a  sin.(a+5)=sin.6-(-sin.a  cos.(a-|-^) 

To   demonstrate   the  truth   of  equation  (10),  we  use  the  last 
figure,  conceiving  the  arc  ^C  to  be  2a,  the  arc  BD  to  be  2b. 


TRIGONOMETRY.  219 

Then  the  arc  BC  will  be  measured  by  (180°— 2(a+5)  );  its  half 
will  therefore  be  measured  by  90° — {a-\-b). 

But,     .     2  sin.(90°— a4-i)=2  cos.(a+i)=^C 

On  this  hypothesis, 

The  chord    ^(7=2  sin.a  )    *,         i>^=2  sin.J  ) 
C7i>=2  cos.a  f  ^^^^'     AB=:^  cos.b  f 

AI)=2,  and  5(7=2  cos.(a4-i) 

Substituting  these  values  in  equation  (2),  we  have 

2  cos.J  2  cos.a=4  cos.(a+^)+2sin.a  2sin.d 

Dividing  and  transposing, 

cos. (a4-i)= cos.a  cos .6 — sin.a  sin. J 

To  demonstrate  equation  (10).  Draw 
the  diameter  AD,  and  on  one  side  of  it 
take  the  arc  AB=2a,  and  on  the  other  side 
take  the  arc  DU=2b.  Join  BD,  AE,  and 
BE.  From  B,  draw  BCF  through  the 
center  of  the  circle ;  then  the  arc  DEF 
=  'the  arc  AB,  and  EF  is  the  difference 
of  the  arcs  AB  and  DE;  it  is  therefore  measured  by  2(a — 6). 

Now,  in  the  quadrilateral  ABDE,  we  have 


AJD'BE=AB'DE+DB'AE 

AB=2  sin.a  )    . ,         J)E=2  sin.b  ) 
BD=2  cos.a  ]  ^^^°'    AE=2  cos.b  \ 

AD=2,  and  BE=2  cos.(a—b) 

These  values,  substituted  in  the  last  equation,  will  give 

4  cos.(a — 6)=2  sin.a  2  sin.5-|-2  cos.a  2  cos.b 

cos. (a — 6)=sin.a  sin.J+cos.a  cos.6 

PROBLEMS    FOR    EXERCISE. 

A 

1.  Show,  yeome/nca/fy,  that  rad.*(rad.4'COS.w4)=2  cos^ — ;  that 

rad.*(rad — cos.A)=2  sin^«-;    that  rad**sin.2-4s=2  sin.-4»cos.<4/ 


220  APPENDIX  TO 

2.  Prove  that  tan.-4+tan.J5= — ^^         {,,  radius  beinff  unity. 

cos.^'cos.^  ^         ^ 

3.  Demonstrate,  geometrically^  that  rad.'sec.2^=tan..4  tan. 2^ 

4.  Show  that  in  any  plane  triangle,  the  base  is  to  the  sum  of  the 
other  two  sides,  as  the  sine  of  half  the  vertical  angle  is  to  the  cosine 
of  half  the  difference  of  the  angles  at  the  base. 

5.  Show  that  the  base  of  a  plane  triangle  is  to  the  difference  of 
the  other  two  sides,  as  the  cosine  of  half  the  vertical  angle  is  to 
the  sine  of  half  the  difference  of  the  angles  at  the  base. 

6.  The  difference  of  two  sides  of  a  triangle,  is  to  the  difference 
of  the  segments  of  a  third  side,  made  by  a  perpendicular  from  the 
opposite  angle,  as  the  sine  of  half  the  vertical  angle  is  to  the  cosine 
of  half  the  difference  of  the  angles  at  the  base ;  required  the 
proof. 

NOTE. 

When  we  give  our  attention  to  the  relations  existing  between  the 
arc  of  a  circle  and  its  sine,  cosine,  and  tangent,  it  becomes  very  desir- 
able to  find  some  law  which  will  invariably  and  unconditionally  nume- 
rically connect  the  arc  with  its  trigonometrical  lines  ;  and  the  object 
has  been  accomplished,  though  not  in  as  elementary  a  manner  as  is 
desirable  for  a  work  like  this. 

In  the  calculus  the  process  is  clear  and  simple ;  but  simple  as  it 
may  be,  the  reader  must  first  understand  the  calculus  before  it  can 
he  even  comprehensible  to  him. 

We  give  the  following  investigation,  independent  of  the  calculus, 
taken  from  the  French  works  of  Legendre,  with  our  own  modifica- 
tions and  illustrations.  By  a  little  careful  study,  any  one  can  thoroughly 
comprehend  it,  who  is  familiar  with  algebraic  equations,  and  under- 
stands the  binomial  theorem, 

LEMMA. 

If  there  he  an  algdrraic  equation  in  which  the  members  consist  of  quan- 
tities, part  real  and  part  imaginary,  then  the  real  quantities  in  the  two 
members  are  equal,  and  the  imaginary  quantities  are  equal. 

N.  B.  Imaginary  quantities  contain  the  factor  J — 1,  and  such 
quantities  are,  emphatically,  imaginary;  they  have  no  real  existence. 


TRIGONOMETRY.  221 

Suppose  we  have  an  equation  in  which  the  sum  of  the  real  quantities 
in  the  first  member  is  represented  by  A ;  and  the  sum  of  the  like 
quantities  in  the  second  member  by  B.  Also,  the  sum  of  the  imagi- 
nary quantities  in  the  first  member,  suppose  represented  by  iSV — 1» 
and  the  sum  of  the  like  quantities  in  the  second  member  by  TJ — 1; 
that  is,  suppose  the  following  equation  to  exist. 

Then,       .  A=J5,  and  SJ^l=zTJ^^ 

If  A  is  not  equal  to  B,  one  must  be  greater  than  the  other  ;  and  as 
they  are  supposed  to  be  real  and  definite  quantities,  their  difference 
must  be  real  and  definite  ;  and,  therefore,  we  can  represent  it  by  the 
definite  quantity  D. 

That  is,  suppose  A  greater  than  B  by  D;  then  the  equation 
becomes 

B+D-]-S^~—l=B-\-Tj^l 
Strike  out  B  from  both  members,  and  transpose  S^ — 1 
Then,         D=Tj^—Sj'^=iT—S)J^l 

That  is,  a  real  quantity  equal  to  an  imaginary  one — a  perfect  absurdity; 
and  this  absurdity  is  in  consequence  of  supposing  A  not  equal  to  B; 
therefore,  we  must  admit  that  A=B, 

It  necessarily  follows  that 


Let  a  represent  any  arc,  the  radius  unity;  then, 

cos.2a-j-sin.2a=l 

Conceive  the  first  member  as  composed  of  the  two  factors, 

co8.a-{-h  sin. a,  and  cos. a — h  sin.o 

The  product  of  these  two  factors,  is 

cos.^a — h^  sin.'a;    and,   by   hypothesis,  this 
product  must  equal  the  first  member  of  the  equation  ;  that  is, 

Dropping  cos. 'a  from  both  members,  there  remains 
—  A^  sin.2a=sin.'a 


222  APPENDIX  TO 

Dividing  by  sin. 'a,  and  changing  signs,  we  have 

h^= — 1,  or  h:=-\-tJ — 1,    which  shows  that    the 
coefficient,  A,  is  imaginary.* 

The  different  powers  of  h  are 

h=+lj^h  A2=— 1,  A5=—l  V^,  h'  =+1,  A5=+J^l,  A«=— 1, 

and  so  on.     Observe  that  all  the  even  powers  of  h  are  rational  quan- 
tities ;  in  short,  units,  with  the  signs  pliis  and  minus  alternating. 

Thus,        .    A2=— 1,  A^=+l,  ;i6=— 1,  A8=-|-l,  and  so  on. 

All  the  odd  powers  are  imaginary,  and  the  signs  alternating. 
If  we  multiply  the  two  similar  factors, 

cos.a-(-A  sin.a 
And,         .        .         .  cos.b-\-h  sin  .6 

Product  will  be,  cos.a  cos.&-{-(sin.a  cos.&-{-cos.a  sm.h^h-^-Ti^Bm.a  sin.ft 
Now  let    A=^ — 1,    and  ^'= — 1;    then  this  product  is 
(cos.  a  C0S.& — sin.a  sin.Z>)-|-(sin.acos.6-|-cos.asin.&)>7 — 1 

Comparing  this  expression  with  equations  (9)  and  (7),  page  141, 
we  perceive  that  it  is  the  same  as 

coB.(ji-\-l)-\-Bm.{a-\-l)^ — 1  ; 
Hence,    (cos.a-f-^  sin.fl)(cos.J-|-A  sin.a)=cos.(a-f-^)+^  sin.(a-|-^) 
In  case  we  give  to  h  its  particular  imaginary  value,  »J — 1 
It  is  very  remarkable  that  the  product  of  these  factors  can  be  found  by 
simply  adding  the  arcs,  which  is  a  property  analagous  to  logarithms. 

If  we  make  a=J  in  the  preceding  equation,  we  have 

(cos.a-f-^  sin.fl)(cos.a-|-^  sin.a)=cos.2o-|-^  sin.2a        (1) 
(cos.a-f-A  sin.a)(cos.2a-f-^  sin.2a)=cos.3a-|-'^  sin. 3a        (2) 
(cos.a-j-A  sin.a)(cos.3a-j-^  sin.3a)=cos.4a-4-A  sin.4a        (3) 
and  so  on. 
The  first  member  of  equation  (1),  is 

(cos.a-|-^  sin.a)2 

*  This  investigation  shows,  also,  that  the  sum  of  any  two  squares  may  be 
regarded  as  the  product  of  two  binomial  factors. 

Thus,       .        .        .  a:24-y2=(;e_|_y^-Zj:)(a;-y^~I) 


TRIGONOMETRY.  233 

The  first  member  of  equation  (2),  is 

(cos.fl-j-A  sin.a)*,  and  so  on.  Therefore,  in 
general,  if  n  is  taken  to  represent  any  entire  number  whatever,  we 
shall  have, 

cos.na+A  sin.«ass=(cos.a-|-A  sin.a)" 

But,        •  (cos.a+A  sin.a)°=cos."«(l4-A  tan.a)" 

sin.a 

Because,         •        .        . =tan.a 

cos.a 

Hence,  .  co8.na-{-h  sm.na=co8.''a(l-\-h  t&n.ay        (4) 

Expanding  the  binomial  in  the  second  member,  we  have 
n — I  n — 1  n — ^2 

Substituting  the  expanded  binomial  in  equation  (4),  it  becomes 

cos.Tia-j-^  sin.7wi= 

n — 1                    n — 1  n — 2 
cos.°a(l-4-TOA  tsLn.a-\-n—^h^  tan.'a-j-TO— r ^-h^  tan.^a,  &c.) 

Calling  to  mind  the  principles  explained  in  the  preceding  lemma, 
and  recollecting  that  all  the  terms  containing  the  odd  powers  of  h  must 
be  imaginary,  and  all  the  other  terms  real,  therefore,  we  may  put 
cos.Tia  equal  to  all  the  real  quantities  in  the  series,  multiplied  by  the 
fa6tor  cos."a;  and  the  imaginary  quantity  h  sin.Twi,  must  be  put  equal  to 
all  the  terms  in  the  series  containing  the  odd  powers  of  h,  and  the 
whole  multiplied  by  the  factor  cos."a. 

But  as  every  term  of  this  equation  will  contain  A,  we  can  divide  by 
h,  and  thus  convert  every  odd  power  into  an  even  power,  and  change 
the  equation  from  imaginary  terms  to  real  terms. 

Thus,  by  equating  the  parts  of  the  preceding  equation,  we  have 

cos.na= 

,    .      ^ — 1  ,  o        «    .      n — 1  n — 2  71—3  , 
cos°a(l+^  -2~      t&nM-\-n  —z ^ j-  h*  tan.''a+  &c.) 

n — 1  n — 2  ,  „        ,          n — 1  n — 2  n — 3 
sin.na=cos."a(7i  tan.a-j-7i  — r o~  ^  tan.^a-j-Ti  — — - 

n — 4 

— r—  h*  tan.*a-|"  ^') 

X 

Put    x=.na.    Then  %=-.     Also  observe  that  h'^=z — l,  and  A*=l, 

and  so  on,  alternately.     Making  these  substitutions,  the   preceding 
equations  become 


APPENDIX  TO 

x'x^-a  tan'a    ,  a?(a?— a)(ar— 2a)(a?— 3a)  tan.<a 
coB^co..»a(l-  -T2-   -^  + T^ ■  ir      ^"J 

/x  tan.a        a:(x — a)(a: — 2a)  tan.'a 

i=C08.-tf  ( ^^ 7-7^^- -- 

\  1      a  1»2'3  a^ 


8in.a:==cos 
x(x — a)(x — 2a)(x — Za)(x — 4a)    tan.*a  \ 

In  these  equations  the  arc  a  may  be  taken  of  any  value  whatever, 

J    1                                             ,,          tan.a    . 
and  when  a  represents  a  very  small  arc,  is  very  near  unity,  and 

is  exactly  unity  when  a=0. 

Also,  when  a=0,  cos.a=l,  and  any  power  of  1  is  1 ;  therefore, 
co8."a=l.  Making  these  substitutions,  the  final  results  will  be, 

"°^-*=l-  Ta+   i^  -  1.2-3.4.6.6      +  *"• 

sin.a:=a:—  -y^g-T  +  ^2^3^.6    "~  1.2-3-4«5.6-7  +  ^*^- 

To  apply  these  equations,  and  show  their  practical  utility  in  the  pri- 
mary computions  for  the  natural  sines  and  cosines,  we  require  the 
natural  sine  and  cosine  of  3°. 

When  radius  is  unity,  the  arc  of  180°  is  3.14169266. 

Therefore,  the  arc  of  3°  is  .062369877. 


Hence, 

a?2 
"~2"~ 

—0.001370733 

And,  . 

x^ 
24  ~ 

4-0.000000313 

Therefore,  from  . 

, 

.     1.000000313 

Take  .        .        . 

. 

.     0.001370733 

cos.a^= 

0.998629580 

x= 

0.062369877 

x' 
"~6"~ 

t  000023923 

x' 

0.000000003 

120"" 

8in.a:=      0.062336957  the  sin.  of  3°. 

In  like  manner  we  may  compute  the  sine  and  cosine  of  any  other 
arc.    But  the  greater  the  arc,  the  slower  the  series  will  converge;  and, 


TRIGONOMETRY.  225 

in  case  of  large  arcs,  a  greater  number  of  terms  must  be  taken  to  obtain 
a  result  of  equal  exactness  ;  the  series,  however,  is  never  used  for 
large  arcs,  but  the  combinations  of  other  formulas  ire  then  used. 
These  formulas  are  more  practical  than  any  other  hitherto  given  for 
the  same  object ;  but  their  theoretical  investigation  is  supposed  to 
require  more  power  than  a  learner  can  at  first  possess. 

15 


226  CONIC    SECTIONS. 


CONIC    SECTIONS. 

DEFINITIONS. 

1.  Conic  Sections  are  the  figures  made  by  a  plane,  cutting  a 
cone. 

2.  There  are  Jive  dififerent  figures  that  can  be  made  by  a  plane 
cutting  a  cone,  namely :  a  triangle,  a  circUy  an  ellipse,  a  parabola, 
and  an  hyperbola. 

Remark.  The  three  last  mentioned  are  commonly  regarded  as 
embracing  the  whole  of  conic  sections ;  but  with  equal  propriety 
the  triangle  and  the  circle  might  be  admitted  into  the  same  family. 
On  the  other  hand  we  may  examine  the  properties  of  the  ellipse, 
the  parabola,  and  the  hyperbola,  in  like  manner  as  we  do  a  triangle 
or  a  circle,  without  any  reference  to  a  cone,  whatever. 

It  is  important  to  study  these  curves  on  account  of  their  exten- 
sive application  to  astronomy  and  other  sciences. 

3.  If  a  plane  cut  a  cone  through  its  vertex,  and  terminate  in 
any  part  of  its  base,  the  section  will  evidently  be  a  triangle. 

4.  If  a  plane  cut  an  upright  cone  parallel  to  its  base,  the  section 
will  be  a  circle. 

6.  If  a  plane  cut  a  cone  obliquely  through  both  sides  of  the 
cone,  the  section  will  represent  a  curve,  called  an  ellipse. 

6.  If  a  plane  cut  a  cone  parallel  to  one  side  of  the  cone,  or 
what  is  the  same  thing,  if  the  cutting  plane  and  the  side  of  the 
cone  make  equal  angles  with  the  base,  then  the  section  will  represent 
a  parabola. 

7.  If  a  plane  cut  a  cone,  making  a  greater 
angle  with  the  base  than  the  side  of  the  cone 
makes,  then  the  section  is  an  hyperbola. 

.8.  And  if  all  the  sides  of  a  cone  be  continued 
through  the  vertex  forming  an  opposite  equal 
cone,  and  the  plane  be  also  continued  to  cut 
the  opposite  cone,  this  latter  section  will  be  the 
opposite  hyperbola  to  the  forfiie^ 


DEFINITIONS. 


227 


9.  The  vertices  of  any  section  are  the  points 
where  the  cutting  plane  meets  the  opposite  sides 
of  the  cone,  or  the  sides  of  the  vertical  trian- 
gular section,  as  A  and  B. 

Hence  the  ellipse,  and  the  opposite  hyperbolas, 
have  each  two  vertices ;  but  the  parabola  only 
one ;  unless  we  consider  the  other  as  at  an 
infinite  distance. 

10.  The  axis,  or  transverse  diameter  of  a  conic  section,  is  the 
line  or  distance  AB  between  the  vertices. 

Hence,  the  axis  of  a  parabola  is  infinite  in  length,  AB  being  only 
a  part  of  it. 


THE    ELLIPSE. 

When  we  know  how  to  describe  a  circle,  we  can  give  a  definition 
of  it ;  and  without  conceiving  it  to  be  a  conic  section,  we  can  go  on 
and  investigate  its  properties.  So  with  the  ellipse.  Wlien  we 
know  how  to  describe  it,  we  can  give  a  definition  of  it,  and  go  on 
and, investigate  its  properties  ;  and  we  shall  do  so  without  conceiving 
it  to  be  a  conic  section. 


PROBLEM. 


To  describe   an  Ellipse, 

Take  any  two  points,  as  F  and  F', 
Take  a  thread,  longer  than  the  dis- 
tance between  F  and  F',  and  fasten 
one  extremity  at  the  point  F,  the  other 
2it  F'.  Then  take  a  pencil  and  put  it 
in  the  loop,  and  move  the  pencil  entirely 
round  the  fixed  points,  keeping  the 
thread  at  equal  tension  in  every  part.  The  pencil  thus  passing 
round  the  points  i'^and  F',  describes  a  curve,  as  is  represented  in 
the  adjoining  figure,  and  it  is  called  an  ellipse  ;  hence  an  ellipse 
may  be  defined  as  on  the  following  page  : 

/      'orTH£  1 


228  CONIC    SECTIONS. 


DEFINITION  S 


1.  An  ellipse  is  a  plane  curve,  confined  by  two  fixed  points  ;  and 
the  sum  of  the  distances  from  any  point  in  the  curve  to  the  fixed 
points,  is  constantly  the  same. 

2.  The  two  fixed  points  are  called  the  foci. 

3.  The  center  is  the  point  C,  the  middle  point  between  the  foci. 

4.  A  diameter  is  a  straight  line  through  the  center,  and  termi- 
nated both  ways  by  the  curve. 

6.  The  extremities  of  a  diameter  are  called  its  vertices. 
Thus,  DD'  is  a  diameter,  and  D  and  D'  are  its  vertices. 

6.  The  major  axis  is  the  diameter  which  passes  through  the/oa. 
Thus,  AA'  is  the  major  axis. 

7.  The  minor  axis  is  the  diameter  at  right  angles  to  the  major 
axis.     Thus  CE  is  the  semi  minor  axis. 

8.  The  distance  between  the  center  and  either  focus  is  called 
the  excerUridty  when  the  semi  major  axis  is  unity. 

That  is,  the  excentricity  is  the  ratio  between  CA  and  CF;  or  it 

CF 
is   -z^;    and,  of   course,  always  less  than  unity.     The  less  the 
L/A 

excentricity,  the  nearer  the  ellipse  approaches  the  circle. 

9.  A  tangent  is  a  straight  Hne  which  meets  the  curve  in  one 
point,  only;  and,  being  produced,  does  not  cut  it. 

10.  An  ordinate  to  a  diameter  is  a  straight  line  drawn  from  any 
point  of  the  curve,  parallel  to  a  tangent^  passing  through  one  of  the 
vertices  of  that  diameter. 

N.  B.  A  diameter  and  its  ordinate  are  not  at  right  angles, 
unless  the  diameter  be  either  the  major  or  minor  axis. 

1 1.  The  points  into  which  a  diameter  is  divided  by  an  ordinate, 
are  called  abscissas. 

1 2.  The  parameter  of  a  diameter  is  the  double  ordinate  which 
passes  tlirough  one  of  the  foci. 

13.  The  parameter  of  the  major  axis  is  called  the  principal 
parameter,  or  latus-rectum.  Thus,  F'G  is  one  half  of  the  principal 
parameter. 

14.  A  subtangent  is  that  part  of  the  axis  produced,  which  is 
included  between  a  tangent  and  the  ordinate  drawn  from  the  pomt 
of  contact. 


THE    ELLIPSE.  229 

PROPOSITION    1.    THEOREM. 

The  maj(yr  axis  is  always  equal  to  the  sum  of  the  two  lines  draton 
from  any  point  in  the  curve  to  the  foci. 

Suppose  the  pencil  at  i>  to  revolve 
along  in  the  loop,  holding  the  threads 
F'D  and  FD  at  equal  tension;  and 
when  D  arrives  at  A,  there  will  be 
two  lines  of  threads  between  F  and  A 
Hence,  the  entire  length  of  the  threads 
will  be  measured  by  F' F-^-^FA. 
Also,  when  D  arrives  at  A'^  the  length  of  the  threads  is  measured 
by  FF'+^F'A'. 

Therefore,       .       FF'-\-'2,FA=FF'-^^F'A' 

Hence,    ....   FA^F'A' 

From  the  expression  FF'^'^.FAy  take  away  FA,  and  add  F'A\ 
and  the  sum  will  not  be  changed,  and  we  have 

FF'+9,FA=:A'F''\-FF'+FA=^A'A 

Hence,  .        .        ,F'D+FD=A'A  Q.  E.  D. 

PROPOSITION    2.     THEOREM. 

The  distance  from  either  focus  to  the  extremity  of  the  minor  axis,  is 
equal  to  half  the  major  axis. 

As  F'Cz=CF  (see  last  %ure),  and  CD  is  at  right  angles  to 

F'F,  therefore,  .         .         .  F'D=FD. 

But,       .        .        ,F'D-{-FD=A'A 

Or,        ...        .  ^D=AA 

Or,        .        .         .        .    FD=  half  A' A,  or  CA.     Q.  E,  D, 

Scholium,     Half  the  minor  axis  is  a  mean  proportional  between 
the  distance  from  either  focus  to  the  principal  vertices. 

In  the  right  angled  triangled  FCD  we  have 

CD^=FI)^~FC^ 
But,      ....    FD=AC 


230 

Therefore, 


Or. 


CONIC    SECTIONS. 

=(  J  C-\-FC){AC—FC) 
=AF'XAF 
AF:  CD=CD:FA' 


PROPOSITION    3.     THEOREM 

Fvery  diameter  is  bisected  in  the  center. 

Let  D  be  any  point  in  the  curve, 
and  C  the  center.  Join  DC,  and 
produce  it.  From  F'  draw  F'D'  parallel 
to  FD;  and  from  F  draw  FD'  parallel 
to  F'D.  The  figure  DFD'F '  is  a  par- 
allelogram by  construction  ;  and  there- 
fore its  opposite  sides  are  equal. 

Hence,  the  sum  of  the  two  sides  F'D'  and  D'F  is  equal  to  F'D 
and  DF;  therefore,  by  definition  1,  the  point  D'  is  in  the  ellipse. 
But  the  two  diagonals  of  a  parallelogram  bisect  each  other  ;  there- 
fore, DG=^  CD',  and  the  diameter  DD'  is  bisected  at  the  center, 
C,  and  DD'  represents  any  diameter.     Therefore,  &c.      Q.  E.  D. 


PROPOSITION    4.    THEOREM. 

A  tangent  to  the  ellipse  makes  equal  angles  with  the  two  straight 
lines  drawn  from  the  point  of  contact  to  the  foci. 

Let  i^  and  i^'  be  the  foci,  and  D 
any  point  in  the  curve.  Join  F'D  and 
FD,  and  produce  F'D  to  JI,  making 
DH=DF,  and  join  FH.  Bisect  FH 
in  T.     Join  TD  and  produce  it  to  t. 

Now  by  theorem  15,  book  1,  the 
angle  FDT=  the  angle  HDT,  and 
HDT=^  its  opposite  vertical  angle,  F'Dt. 

Therefore,        .        .        FDT=^F'Dt 

It  now  remains  to  be  shown  that  71  is  a  tangent,  and  only  meets 
the  curve  at  the  point  D. 


THBELLIP&E.  231 

If  possible,  let  it  meet  the  curve  in  some  other  point,  as  t,  and 
join  Ft,  tJI,  and  F't. 

By  theorem  15,  book  1,      Ft=tH 

To  each  of  these  add  F't; 

Then,         .        .  F't-^-tH^^F't^Fi 

But  F't-\-tH  are,  together,  greater  thanJ^^,  because  a  straight 
line  is  the  shortest  distance  between  two  points  ;  that  is,  F't-\-Fiy 
the  two  lines  from  the  foci,  are,  together,  greater  than  FH,  or 
greater  than  F'DArFD;  therefore,  the  point  t  is  without  the 
ellipse,  and  t  is  any  point  in  the  line  Tt,  except  D;  therefore,  Tt  is 
a  tangent,  touching  the  ellipse  at  D,  and  it  makes  equal  angles 
with  the  lines  drawn  from  the  point  of  contact  to  the  foci. 

q.  E.  D. 

Cor.  The  tangents  at  the  vertices  of  either  axis  are  perpendicular 
to  that  axis ;  and  as  the  ordinates  are  parallel  to  the  tangents,  it 
follows  that  all  ordinates  to  the  major  or  minor  axis  must  cut  one 
axis  at  right  angles,  and  be  parallel  to  the  other  axis. 

Scholium,  Axij  point  in  the  curve  may  be  considered  as  a  point 
in  a  tangent  to  the  curve  at  that  point. 

It  is  found  by  experiment  that  light,  heat,  and  sound,  when  they 
approach  to,  are  reflected  off,  from  any  surface  at  equal  angles;  that 
is,  any  and  every  single  ray  makes  the  angle  of  reflection  equal  to 
the  angle  of  incidence. 

Therefore,  if  a  light  is  placed  at  one  focus  of  an  ellipse,  and  the 
sides  a  reflecting  surface,  the  reflections  will  concentrate  at  the 
other  focus.  If  the  sides  of  a  room  be  elliptical,  and  a  stove  is 
placed  at  one  focus,  it  will  concentrate  heat  at  the  other. 

Whispering  galleries  are  made  on  this  principle,  and  all  theaters 
and  large  assembly  rooms  should  more  or  less  approximate  to  this 
figure.  The  concentration  of  the  rays  of  heat  from  one  of  these 
points  to  the  other,  is  the  reason  why  they  are  called  the  foci,  or 
burning  points. 

PROPOSITION    5.    THEOREM. 

Tangents  to  the  ellipse,  at  the  vertices  of  the  diameter,  are  parallel  to 
one  another. 


232 


CONIC    SECTIONS. 


^  I)' 


Let  DB'  be  the  diameter,  and  F'  and 
F  the  foci.  Join  F'D,  F'D',  FD,  and 
FD\ 

Draw  the  tangents,  7^  and  Ss,  one 
through  the  point  i>,  the  other  through 
the  point  D',  These  tangents  will  be 
parallel. 

By  proposition  3,  F'D'FD  is  a  parallelogram,  and  the  angle 
F'D'F  is  equal  to  its  opposite  angle,  F'DF, 

But  the  sum  of  all  the  angles  that  can  be  made  on  one  side  of  a 
line,  is  equal  to  two  right  angles. 

Therefore,  by  leaving  out  the  equal  angles  which  form  the 
opposite  angles  of  the  parallelogram,  we  have 

8D'F'-^SD'F'=tDF'-\-TDF, 

But,  by  proposition  4,  sD'F'=SD'F;  therefore,  their  sum  is 
double  of  either  one  of  them,  and  the  above  equation  may  be 
changed  to         .        .     ^SD'F='itDF 

Or,        .        .        .       SD'F=tDF' 

But  BF'  and  B'Fare  parallel ;  therefore,  SB'F  and  tDF'  are, 
in  effect,  alternate  angles,  showing  that  Tt  and  Ss  are  parallel. 

Q.  F.  D, 

Cor.  If  tangents  be  drawn  through  the  vertices  of  any  two 
conjugate  diameters,  they  will  form  a  parallelogram  circumscribing 
the  ellipse. 


PROPOSITIONS     THEOREM. 

If y  from  the  vertex  of  any  diameter,  straight  lines  are  drawn  through 
thefody  meeting  the  conjugate  diameter y  the  part  intercepted  by  the 
conjugatey  is  eqvxd  to  half  the  vnajor  axis. 

Let  DD'  be  the  diameter,  and  Tt 
the  tangent.  Draw  FF'  parallel  to 
Tt.  Join  F'D  and  DF,  and  produce 
J)F  to  F;  and  from  F  draw  FO 
parallel  to  FF'  or  Tt. 

Now,  by  reason  of   the  parallels. 


THE    ELLIPSE.  233 

we  have  the  following  equations  among  the  angles. 

TDF=DFa  5       ^  '  TDF=^DKH  S 
But,  by  proposition  4,       tD  0=  TDF 
Therefore,  by  equality,    DQF^BFQ 
And,        .        .        .      DHK=DKH 

Hence,  the  triangle  DQF  is  isosceles;  also,  the  triangle  DHK 
is  isosceles.    Whence,    .  DG=DF,  and  Dff=DK 

Because  ffO  is  parallel  to  FO,  and  F'C=CF, 

Therefore,        .        .        F'H=^HQ 

Add         .        .  DF=DQ 

F'H-{-DF=DH 

But  the  sum  of  the  lines  in  both  members  of  this  equation  is 
F'D-^DF,  which  is  equal  to  the  major  axis  of  the  ellipse  ; 
therefore,  either  member  is  half  the  major  axis ;  that  is,  DJI,  or 
its  equal,  DIT,  is  each  equal  to  half  the  major  axis.     Q.  F.  D. 

PROPOSITION    7.     THEOREM. 

Perpendiculars  from  the  foci  of  an  ellipse  upon  a  tangent,  meet  the 
tangent  in  the  circumference  of  a  circle,  whose  diameter  is  the  major 
axis. 

Let  F'F  be  the  foci,  G  the  center,  and  D  a  point  in  the  ellipse, 
through  which  passes  the  tangent  Tt.  Join  F'D  and  FD,  and 
produce  F'D  to  H,  making  DH—FD,  and  produce  FD  to  (7, 
making  DQ^F'B,  Then  F'H  and  FQ  are  each  equal  to  the 
major  axis,  A' A, 

Join  FHy  meeting  the  tangent  in  Ty  and  join  F'  O,  meeting  it 
in  t.     Draw  the  dotted  lines,  GT  and  Gt. 

By  proposition  4,  the  angle  FDT=  the  angle  F'Dt;  and  observ- 
ing that  opposite  vertical  angles  are  equal,  therefore,  the  four  angles 
formed  by  lines  crossing  at  i>,  are  all  equal. 

The  triangles  DF'  G  and  DHF  are  isosceles  by  construction,  and 
as  their  vertical  angles  at  D  are  bisected  by  the  line  Tt,  therefore, 
F't^tG,  and  FT=  TH. 


234  CONIC    SECTIONS. 

Comparing  the  triangles  F'OF  and 
F'Ct,  we  find  FC  equals  the  half  of 
F'F,  and  F't  the  half  of  FCf;  therefore, 
Ct  is  the  half  of  FG.  But  A'A=FO; 
hence,  Ct=iA'A=OA, 

Comparing  the  triangles  FF'H  and 
FCT,  we  find  the  sides  FH  and  FF' 
cut  proportionally  in  T  and  0;  therefore, 
they  are  equiangular  and  similar,  and 
CT  is  parallel  to  F'H,  and  equal  to  half  of  it.  That  is,  CT  is, 
equal  to  CA;  and  CAy  CT,  and  Ct,  are  all  equal ;  and  hence  a 
circle  described  from  the  center,  (7,  at  the  distance  of  CA,  will  pass 
through  the  points  T  and  t.     Therefore,  perpendiculars,  &c. 

Q.  E.  D. 

PROPOSITION    8.     THEOREM. 

The  product  of  the  perpendicvlars  from  the  foci  upon  a  tangent,  is 
equal  to  tJte  square  of  half  the  minor  aocis. 

Produce  TC  and  GF'  (see  figure  to  the  last  proposition),  and 
they  will  meet  in  the  circle,  at  S;  for  FT  and  F't  are  both  per- 
pendicular to  the  same  line,  Tt;  they  are,  therefore,  parallel ;  and 
the  two  triangles  CFT  and  CF'  S,  having  a  side,  FC,  of  the  one, 
equal  to  CF',  of  the  other,  and  their  respective  angles  equal, 
therefore  CS=CT,  and  S  is  in  ihe  circle,  and  SF'=FT. 

Now,  as  A' A  and  St  are  two  lines  that  intersect  each  other  in 
a  circle,  therefore,  (th.l7,  b.  3) 

SF'XF't=-A'F'XF'A 
FTXF't=A'F'XF'A 

But,  by  the  scholium  to  proposition  2,  it  is  shown  that 

A'F'XF'A=^  the  square  of  half  the  minor  axis. 

Hence,     .         .       FTX  F't-=  the  square  of  half  the  minor  axis. 

Therefore,  the  product,  &c.     Q.  E.  D. 

Cot,  The  two  triangles,  FTD  and  F'tB,  are  similar,  and  from 
them  we  have  .  TF  :  F't=FD  :  DF';  that  is,  perpendiculars 
let  fall  from  the  foci  upon  a  tangent,  are  to  each  other  as  the  distances 
of  the  point  of  contact  from  the  foci. 


THE    ELLIPSE 


235 


PROPOSITION    9.    PROBLEM. 

Oiven  the  major  aans  and  the  distance  between  the  fod  of  any  ellipse^ 
to  find  the  relation  between  an  abscissa  of  the  major  axis  and  its  cor- 
responding ordinate. 

Let  F'  and  F  be  the  foci,  C  the 
center,  and  put  CF',  or  CF=c,  and 
CA=A.  Then  F'J)=A,  and  in  the 
triangle  F' DO  or  FDO,  if  the  hypo- 
tenuse FJ)  and  FC  are  both  known, 
then  DC  is  known;  therefore,  we  may 
put  CD=B,  and  consider  A,  By  and 
c,  known  quantities. 

Take  any  point  on  the  major  axis,  as  t,  and  draw  tP  at  right 
angles  to  A' A, 

Measuring  from  the  point  A',  A't  is  the  abscissa,  and  tP  is  the 
corresponding  ordinate. 

The  problem  requires  us  to  find  the  mathematical  relation 
between  these  two  hues.  We  can  find  it  by  the  aid  of  the  two 
right  angled  triangles  F'tP  and  FtP, 


Put        .        .      A't=x,  and  tP=y 

Then      .        .    F't=A't—A'F'=x--(A—c)= 

=x+c^A 

And       .        .       Fi=:A't-^A'F=x^(A+c)= 

-x    c    A 

Put        .        .   F'P=r,SindF'P=r' 

Then,     .          F'P+FP=r'+r=2A 

0) 

In  the  triangle  FPt  we  have 

(x-[-c^Ay+f==r'' 

(2) 

In  the  triangle  FPt  we  have 

(x^^Ay+y'=r^ 

(3) 

By  subtracting  (3)  from  (2),  expanding  and  reducing,  we 

4ca^— 4c^=r'^— r' 

(4) 

Or,         .        .         .      4c(a:— ^)=(/-i-r)(/— 

r)(6) 

236  CONIC    SECTIONS. 

But  tlie  first  factor  in  the  second  member  of  equation  (6)  is 
equal  to  2A;  hence  we  have 

r'^^±.(^-.A)  (6) 

But,        .        .        .       r'+r=2A  (7) 

By  adding  (6)  and  (7),  then  dividing  by  2,  and  then  subtracting 
(6)  from  (7),  and  dividing  by  2,  we  have  the  two  following 
equations : 

f=A+^(x-A)       (8) 

r=^-j{'-^)       (9) 

It  should  be  observed  that  equations  (8)  and  (9)  are  expressions 
for  lines,  one  of  which  is  called  radius  rector  in  astronomy. 

By  squaring  equation  (9),  and  comparing  it  with  equation  (3), 
equating  the  two  values  of  r^  we  shall  then  have 

ar2+c^^^2— 2car— 2^a;+  9.cA+y^= 
A^—Zc{x--A)'^~(x—AY 

Or,         .  ar^+c*— 2^+y'=^'(^— 2a:^+^') 

Or,      A''x^-\'c'A^—2A^x+Ahf  ^=:=(^x'—2c'xA+c'A^ 

Or,         .  Ay+(A'—c'y=(A^--c')2Ax 

Observing  that  A^ — c^=B\  the  square  of  the  semi  minor  axis, 
and  substituting  this  value,  the  preceding  equation  becomes 

u4y4-^a^=2^^2a; 
Hence,   .        •        .        .       f=:~^{2Ax—x')        (10) 

Or         ....        y^^^i-^J^Ax—x"    (11)  ^ 

We  cannot  reduce  this  equation  to  lower  terms,  or  condense  it 
to  a  more  simple  form ;  and,  therefore,  it  must  rest  as  the  final 
result;  and,  in  the  language  of  analytical  geometry,  it  is  called 
the  equation  of  the  ellipse. 


THE    ELLIPSE.  237 

Any  definite  value  may  be  assigned  to  ic,  not  greater  than  2^, 
and  when  any  particular  value  is  assigned,  the  equation  will  give 
the  corresponding  value  of  the  ordinate,  y,  and  as  y  has  the  double 
sign,  it  shows  that  y  may  be  drawn  both  above  and  below  A' A,  or 
shows  that  the  curve  is  symmetrical  on  both  sides  of  A' A. 

Now  let  us  examine  the  result  when  particular  values  are  given 
to  X.  At  the  point  A'  x=0;  and  this  value  of  x  put  in  the  equa- 
tion, gives  y=0;  obviously  the  proper  result.  Again,  suppose 
x^=2Af  and  this  value  of  x  put  in  the  equation,  gives 

y=±^  V4Z2^T'=±^  X  0 

That  is,  y=0,  for  that  point,  also. 

If  we  suppose  x=3A,  y  will  come  out  imaginary;  showing  that 
there  is  no  real  value  to  y  beyond  the  point  A;  and  in  this  way 
imaginary  equations  have  real  practical  utility. 

If  we  suppose  a;=-4,  then  y  will  become  CD=^B. 

If  we  make  AF'-=.x,  then  x=^A — c;  and  this  value  put  in  the 

equation,  gives    .        y = dz-j  J (  %A — x)  ( A — c) 


By  the  definition,  the  double  ordinate  from  either  focus,  is  called 
the  'parameter;  and  we  perceive  by  this  equation  that  the  semi 
parameter  is  the  third  proportional  to  the  major  and  minor  axes ; 

For,  .  .  A\  B=^B  :  y;  a  proportion  that  gives  the 
preceding  equation. 

It  is  sometimes  most  convenient  to  take  C,  the  center  of  the 
ellipse,  for  the  zero  point,  in  place  of  the  point  A\  one  extremity 
of  the  major  axis. 

If  we  make  this  change,  it  will  cause  no  changes  in  the  ordinate 
y,  but  X,  in  the  equation  for  the  ellipse,  must  be  diminished  by  A; 
and  X,  a  measure  from  that  point,  can  never  be  greater  than  A, 
but  it  can  have  the  double  sign  plus  or  minus.  At  the  point  A\  x 
will  be  equal  to  minus  A,  and  at  the  other  extremity  of  the  major 
axis,  X  will  be  equal  U>plus  A. 

To  change  the  equation   y^=^-^^{^Ax — a?)    into  its  equivalent 


238  •      CONIC    SECTIONS. 

expression,  when  the  origin  of  x  is  changed  from  A'  to  (7,  we  must 
put  X — A=x'.  Hence,  x  and  x'  designate  the  sanie  point  on  the 
axis ;  and  if  a;  is  less  than  A,  then  x'  is  negative. 

If        .       X — A=zx',  then  x=^A-i-z' 

(2Ax'-^^)=(^A'^x)x=(A--x')(A'\-x')=A^—^'^ 

Hence,         f^±lA-^x-)=B^-^^-^ 

Or,        .        .      AY-\~Bx'^^A''I^ 

We  may  omit  the  accent  of  x,  for  x,  or  x' ,  is  only  a  different 
symbol  for  any  point  on  the  major  axis  corresponding  to  the  ordinate 
y.  The  accent  was  only  taken  to  avoid  confusion  while  changing 
the  zero  point ;  therefore,  the  following  equation  is  the  equation  for 
the  ellipse,  the  zero  point  being  the  center. 

'     Ay-\'B'x'=A^J3' 

In  case  -4s=J5,  the  ellipse  becomes  a  circle,  and  the  equation 

becomes        •        .    Ah^-\-A^3^=A* 

Or,   .        .        .        .   f+a^=A* 

This  last  equation  is  obviously  the  equation  of  the  circle,  y  being 
the  sine  of  any  arc,  x  its  cosine,  and  A  the  radius. 

The  change  in  the  zero  point  from  the  vertex  of  the  major  axis 
to  the  center,  changes  equations  (8)  and  (9)  into 

r^A r 


ex  ex 

Or,  without  the  accent,  r'=^+-j ,   and  r=A — j 


PROPOSITION    10.     THEOREM. 

The  squares  of  the  ordinates  of  the  major  axis  are  to  each  other  ob 
the  rectangles  of  their  corresponding  abscissas. 


THE    ELLIPSE. 

Let  y  be  any  ordinate,  and  x  its  corres- 
ponding abscissa.  Then,  by  the  last  pro- 
position, we  shall  have 

Let  y'  be  any  other  ordinate,  and  a;'  its 
corresponding  abscissa,  and  by  the  same  proposition  we  must  have 

Dividing  one  of  these  equations  by  the  other,  omitting  common 
factors  in  the  numerator  and  denominator  of  the  second  member  of 
the  new  equation,  we  have 

y^  _(2^ — x)x 

Hence,  .     y*  :  y'^=(^A—x)x  :  {%A—x)x' 

By  simply  inspecting  the  figure,  we  cannot  fail  to  perceive  that 
(2-4 — x),  and  Xy  are  the  abscissas  corresponding  to  the  ordinate  y, 
and  {^A — x')  and  x'y  are  the  two  corresponding  to  y'.  Therefore, 
the  squares  of  the  ordinates,  &c.     Q.  E.  D. 


PROPOSITION    11.     THEOREM. 

J^  a  circle  he  described  on  the  major  axis  of  an  ellipse,  and  any 
ordinate  be  dravm  common  to  both  the  circle  arul  the  ellipse,  the  ordinate 
corresponding  to  the  circle  is  to  the  part  corresponding  to  the  ellipse  as 
the  major  axis  of  the  ellipse  is  to  its  minor  axis. 

On  A' A  (see  figure  to  last  proposition),  as  a  diameter,  describe 
a  circle.  Draw  any  ordinate,  as  OH.  The  part  DH  is  y,  of  the 
last  proposition. 

The  proportion  in  the  last  proposition  is  true,  and  y  and  y'  may 
be  any  two  ordinates,  whatever.  And  now  suppose  y'  represents 
the  semi  minor  axis  ;  then  x'  will  equal  A,  and  9. A — x'=^A, 
Taking  this  hypothesis,  the  proportion  referred  to  becomes 

f  :  B''={<itA--x)x  :  A^ 


240  CONIC    SECTIONS. 

Changing  the  means,  and  observing  that 

(2A—x)x=  GH^        (th.  17,  b.  3,  schoHum.) 

We  have,     .        f  :  QE^z=.B  :  A^ 

Taking  extremes  for  means,  and  extracting  the  square  root  of 
every  term,  we  have 

QH:y=zA:B  Q.  E.  D. 


PROPOSITION     12.     THEOREM. 

The  area  of  an  ellipse  is  a  mean  proportional  between  two  circles — 
the  one  described  on  the  minor y  and  the  other  on  the  major  axis. 

On  the  major  axis  describe  a  circle,  as 
in  the  figure,  and  draw  Off,  any  ordinate, 
and  conceive  it  to  be  a  broad  lin£,  covering 
portions  of  both  the  circle  and  the  ellipse. 

By  the  last  proposition  we  have 

A  :B=OH  :y 

=  Off':y' 

=  OB":y" 

That  is,  Off',  y';  Off",  y'\  &c.,  are  other  ordinates,  all  in  the 
same  proportion  of  -4  to  B:  and  thus  we  can  conceive  the  whole 
areas  of  both  circle  and  ellipse,  made  up  of  ordinates,  each  and 
all  of  which  are  in  the  proportion  of  A  to  B,  Now,  by  applying 
theorem  7,  book  2,  we  have 

A  :  ^=  QE-\-  Gff\  &c.  :  y^-y\  &c. 
That  is,         .        A  \  B=  area  circle  :  area  ellipse 
But  the  area  of  the  circle  on  the  major  axis,  is  TtA^  (th.  1,  b.  5.) 
Substituting  this,  and  the  proportion  becomes 

A  :  B=rtA^  :  area  ellipse. 

Or,     .         .  area  ellipse =rt^^ 

Which  is  the  mean  proportional  between  {r(A^)  and  (ytJB^),  the 


THE    ELLIPSE.  241 

expressions   for  the  areas  of  the  two  circles,  one  on  the  major 
diameter,  and  the  other  on  the  minor  diameter.     Q.  E.  D. 

Scholium,  Hence  the  rule  in  mensuration  to  find  the  area  of  an 
ellipse. 

Rule.  Multiply  together  the  semi  major  and  semi  minor  axes, 
and  multiply  that  product  by  3.1416, 


PROPOSITION     13.     THEOREM. 

J^  a  cone  he  cut  by  a  plane,  making  an  angle  with  the  base  less  than 
thai  mxide  by  the  side  of  the  cone,  the  section  is  an  ellipse. 

Let  von,  be  a  plane  passing  through  the  axis  of  a  cone,  Anmo, 
another  plane  perpendicular  to  the  former,  cutting  both  sides  of  the 
cone  but  not  parallel  with  the  base  of  the  cone,  then  the  figure 
AnmA'o,  will  be  an  ellipse,  AA'  being  its  major  axis. 

Take  any  point,  t,  and  in  the  plane  AnA'  draw  tn,  at  right  angles 
to  AA'.  and  as  the  plane  AnA'  is  perpendicular  to  the  plane  VOB, 
in  is  at  right  angles  to  all  lines  that  can  be 
drawn  in  the  plane  VGIf,  from  the  point 
/  ;  therefore,  tn  is  at  right  angles  to  J3D, 
Through  the  point  t,  conceive  £D 
drawn  parallel  to  the  base  of  the  cone, 
and  it  will  be  a  diameter  to  a  circular 
section  of  the  cone  passing  through  the 
point  n. 

In  the  same  manner  take  any  other 
point  in  ^^'  as  /,  and  draw  Im  at  right 
angles  to  A' A,  &o  ;  and  Gmll  will  be 
a  circular  section  passing  through  the  point  m. 

Now  by  the  similar  triangles  AtD,  AlH,  A'lQ-,  A'  tB,  we  have 

At :  Al=zDt :  ffl 

A't:A'l=Bt\Gl 

By  multiplying  these  proportions  together  (th.  II,  b.  2),  term, 
by  term,  we  have 

AfA't :  Al'A'l=Df£t :  HI*  01 
16 


fi42  CONIC    SECTIONS. 

But  by  reason  of  the  circle  Bnl),       Bt'Dt=-tir  ^ 

^  [  (th.  17,  b.  2). 

"  circle  Gmff,    Hl'Ol^hir) 

Hence,         .         .  AfA't:Al*A'l=tn'^  :bn^ 

This  last  proportion  shows  the  same  property  as  demonstrated  in 
Proposition  10 ;  therefore,  this  section  of  the  cone  is  an  ellipse. 

Q.  E,  D 

Scholium,  Hence  the  propriety  of  calling  an  ellipse  a  conic 
section. 

PROPOSITION     II.     PROBLEM. 

Oiven  the  major  ctxis,  the  distance  between  the  center  and  eiih&r  focus 
of  an  ellipse y  and  the  angle  made  between  the  major  axis  and  a  radius 
drawn  from  either  focus  to  any  point  in  the  ellipse  to  find  an  expression 
for  that  radius. 

Let  j^  be  a  focus,  and  FP  any  radius, 
and  put  the  angle  PFD=v. 

From  proposition  9,  equation  (m)  we 
find  that 

FP=r^A+~ 
A 

an  equation  in  which  A  represents  the  semi  major  axis,  c  the  dis- 
tance FGf  and  x  the  distance  CD, 

Now  by  trigonometry  we  have 

I :  cos.v=r :  c-^x 

Whence,         .         .         .  x=r  cos.t; — c 

Substituting  this  value  of  x  in  the  equation  for  the  radiiir,  we 
have 

.  ,  cr  cos.v — c* 

Ar=A^-\-cr  cos.t; — c^ 
Hence,  .        {A — c  cos.v)r=^^ — c^ 


Or, 


A^-^ 


A — c  cos.v 


THE    ELLIPSE 


ilS48 


This  equation  shows  the  value  of  ^  in  known  quantities,  and  of 
course  it  is  the  expression  required. 

Scholium.  The  excentricity  of  an  ellipse  is  the  distance  from  the 
center  to  either  focus,  when  the  semi  major  axis  is  taken  as  unity. 
Designate  the  excentricity  by  e,  then  1 :  e=-4 :  c 

Hence, c=eA 

Substituting  this  value  of  c  in  the  preceding  equation,  we  have 


r=: 


A — eA  cos.v     1- 


cos.v 


This  equation  gives  an  expression  for  FPy  when  the  angle  PFD 
is  less  than  90° ;  when  greater  than  90°,  the  expression  is 

A{U-fl 
l-f-e  cos.v 


PROPOSITION     15,     PROBLEM. 

Given  the  relative  values  of  three  difereni  radii,  draicn  from,  the 
focus  of  an  ellipse,  together  with  the  angles  between  them,  to  find  the 
relative  major  axis  of  the  ellipse, -^the  excentricity,  and  the  position  of 
the  major  axis,  or  its  angle  from  one  of  the  given  radii. 

Let  r,  r',  and  r",  represent  the  three 
given  radii,  the  angle  between  r  and  r' 
equal  m,  and  between  r  and  r"  equal  n. 
The  angle  between  the  radius  r  and  the 
major  axis  is  supposed  to  be  unknown, 
and  we  therefore,  call  it  x. 

From  the  last  proposition,  we  have 


1 — e  COS.  a? 


(1) 


1 — €cos(x-{-m)        ^   ^ 
1 — ecos.^x-f-n)       ^  ^ 


244  CONIC    SECTIONS. 

Equating  ^(1 — c^)  obtained  from  (1)  and  (2),  and  we  have 
r — re  cosuc=r' — r'e  cos.(a:4-»w) 


Or,        .  e= -, 1—,—.  (4) 

rcos^ — r  cos.(a;i-m)  ^   ' 

In  like  manner  from  (1)  and  (3), 

T — recos^=r" — r"e  cos.(a;+») 


(5) 


r  cosuc — r"  Q.Q9,.(x-\-n) 
Equating  (4)  and  (6),  we  have 


r  cos.ar — r'cos.(a;+m)     r  cos.a; — r"cos.(a:+w) 
T  cos.a; — r'cos.(a;+77i) 


r — r"     r  cos-a; — r"cos.(a:4-w) 

r  cos.a; — /cos.ar  cos.«i+/sin.a;  sin.m 

r  cos.a; — r"cos.a:  cos.»4-r"sin.a;  sin.» 

r — r'cos.m+r'sin.m  tan.a; 


'"cos.w-|-r"sin.TO  tan.a; 

For  the  sake  of  perspicuity  and   brevity,  put  r — r'=dy 
And  r — r"=d'.     The  known  quantity  r — r'cos.w=a. 

And   r — r"cos.w=6.     Then  the  preceding  equation  becomes, 

d     o-l-r'sin.m  tan.aj 

d''^b-\-r"sm.n  tan.a? 

dh-\'dr"sm.n  iQX\..x=ad  '+cf  Vsin.m  tan.a? 

[dr"8m.n — d'rsin.m)iaTi.x=ad' — dh 

ad'—dh 
dr  sm.w — dr^ivi.m 

The  value  of  x  found  by  this  last  equation,  determines  tld 
position  of  the  major  axis. 

Having  x,  equation  (4)  or  (5),  will  give  the  excentricity  e. 
Equations  (1),  (2),  and  (3),  contain  Ay  the  semi  major  axis  as  a 
common  factor t  it  does  not  therefore  affect  the  relative  values  of  r,  r', 
and  /',  and  as  A  disappears  in  the  subsequent  part  of  the  investi- 


THE    ELLIPSE.  245 

gation,  it  shows  that  the  angle  x  and  the  eccentricity  e,  are  entirely 
independent  of  the  magnitude  of  the  ellipse  ;  they  only  determine 
its  figure.  To  apply  the  preceding  formulas,  we  propose  the 
following 

EXAMPLE. 

On  the  first  day  of  Augiist  1 846,  an  astronomer  observed  the  sun's 
longitude  to  be  128°  47'  31",  and  by  comparing  this  observation  with 
observations  made  on  the  previotcs  and  subsequent  days^  he  found  its 
motion  in  longitude  was  then  at  the  rate  of  57'  24"  9  per  day.  By 
like  observations,  made  on  the  first  of  September,  he  determined  the 
iun*s  longitude  to  be  158°  37'  46",  and  its  mean  daily  motion  for 
that  time  58'  6"  6  ;  and  at  a  third  time,  on  the  10th  of  October,  the 
observed  longitude  was  196°  48'  4",  and  mean  daily  motion  59'  22"  9. 
From  these  data  is  required  the  longitude  of  the  solar  apogee,  and 
the  excentricity  of  the  apparent  solar  orbit. 

It  is  demonstrated  in  astronomy,  that  the  relative  distances  to  the 
sun,  when  the  earth  is  in  diflferent  parts  of  its  orbit,  must  be  to 
each  other  inversely  as  the  square  root  of  the  sun's  apparent  angu- 
lar motion  at  the  several  points ;  therefore,  (r)^  (>'')^  and  (/')', 
must  be  in  proportion  to 

•     1  1  ,1 

;,  and 


57' 24"  9'     58' 6"  6'         59' 22"  9 
Or  as  the  numbers, 

.,  and 


3444.9'       3486.6*  3562.9* 

Multiply  by  3562.9  and  the  proportion  will  not  be  changed,  and 
we  may  put 

/  3562,9  \  4  ,      /  3562.9  \  A  ^     „     ^ 

'•={3444:9)    •        '•-(3486-:6)    '    ^"'^'•='- 

By  the  aid  of  logarithms,  we  soon  find 

r=1.016982  r'= 1.010867  and  r"=l. 

Hence,  r-—r'=cf=0.006 125,       r— /'=<;' =0.0 169  8  2 

158°  37'  46"      196°  48'  4" 
128  47  31       128  47  31 


m=  29  50  15    w=  68   0  33 


fm  CONIC    SECTIONS. 

To  correspond  with  the  formulas,  we  must  take  the  natural  sine 
and  cosine  of  m  and  n, 

m=29°  60'  15"  sin.  .497542  .  cosine  .867440 

«=68     0  33    sin.  .927238  .  cosine  .374472 

^'cos.m=a=0.140172 

'cos.w=5=0.642510 

a<?'=(0.140172)(0.016982)=0.0023796 
W=(0.64251)(0.006125)=0.0039358     . 
c?V'sin.m=0.0085405 
d'/'sin.w=0.0056793 
ad'---bd  db—ad' 


tan.{p= 


dr"^m.n — G?V'sin.m    c?V'sin.m — dr'^m.n 
.0015562     155.62 


.0028612     286.12 


This  numerical  result  corresponds  to  radius  unity  ;  to  compare  it 
with  our  tables  and  take  out  the  arc,  we  must  take  out  the  loga- 
rithm of  the  numerator,  increase  its  index  by  10,  and  subtract  the 
logarithm  of  the  denominator. 

Thus,         .        156.62  log.         .         12.192080 
286.12  log.         .  2.456548 

x=  30°  23'  40"  tan.        9.735532 
From,        ....  128°  47'  31" 

Take,  a?     .         .        .         .  28°  32'  24" 


Longitude  of  the  apogee,    .  100  14  67 

The  true  longitude  at  that  time  was  99°     40'. 

The  result  of  any  one  set  of  observations,  are  but  first  approxi- 
mations, of  course ;  but  we  did  not  adduce  this  example  to  teach 
astronomy,  but  to  teach  the  properties  of  the  ellipse. 

To  find  the  excentricity,  we  apply  equation  (5),  observing  that 
r"cos.(a;+w)  must  be  subtracted,  but  when  {x-\-n)  is  greater  than 


THE    ELLIPSE.  247 

90°  (as  it  is  in  this  case)  it  becomes  negative,  and  substracting  a 
negative  quantity  gives  an  increase, 

r— ^"  _  '0^6982        .016982 

Thus,  e=^  cos.a;— r"  cos.(x-\-n)~"M7T^i''~^0l 

This  ^ves     ^=0.01696  ;    its  true  value  is,     0.01678. 
Our  value  of  re  is  a  little  too  small  which  is  the  principal  cause 
of  the  difference. 


THE    PARABOLA. 

DEFINITIONS. 

1 .  A  parabola  is  a  plane  curve,  every  point  of  which  is  equally 
distant  from  a  fixed  point  and  a  given  straight  line. 

2.  The  given  point  is  called  the/oct^s,  and  the  given  line  is  called 
the  directrix. 

To  describe  a  parabola* 

Let  CD  be  the  given  line,  and  F  a  given 
point.  Take  a  square,  as  DBO^  and  to 
one  ,side  of  it,  QB,  attach  a  thread,  and 
let  the  thread  be  of  the  same  length  as  the 
side  GB  of  the  square.  Fasten  one  end  of 
the  thread  at  the  point  G,  the  other  end  at  F, 

Put  the  other  side  of  the  square  against  the  given  line,  CD,  and 
with  a  pencil,  P,  in  the  thread,  bring  the  thread  up  to  the  side 
of  the  square.  Slide  one  side  of  the  square  along  the  line  CD, 
and  at  the  same  time  keep  the  thread  close  against  the  other  side, 
permitting  the  thread  to  slide  round  the  pencil  P.  As  the  side  of 
the  square,  BD,  is  moved  along  the  line  CDy  the  pencil  will  describe 
the  curve  represented  as  passing  through  the  points  V  and  P, 
OP-^-PF^z  the  thread 
OP+PB=  the  thread 

By  subtraction    PF—PB=0  or  PF=PB 

This  result  is  true  at  any  and  every  position  of  the  point  P;  that 
is,  it  is  true  for  every  point  on  the  curve  corresponding  to  definition  1. 
Hence,        .        .         FV=:VB 


348  CONIC    SECTIONS. 

If  the  square  be  turned  over  and  moved  in  the  opposite  direction, 
the  other  part  of  the  parabola,  the  other  side  of  the  line  FH^  may 
be  described. 

3.  A  diameter  to  a  parabola  is  a  straight  line  drawn  through  any 
point  of  the  curve  'perpendicular  to  the  directrix.  Thus,  the  line 
HF  is  a  diameter;  also,  -56''  is  a  diameter;  and  all  diameters  are 
parallel  to  one  another. 

4.  The  point  in  which  the  diameter  cuts  the  curve,  is  called  the 
vertex  of  that  diameter. 

5.  The  diameter  which  passes  through  the  focus,  is  called  the 
principal  diameter,  and  sometimes  it  is  called  the.  am  of  the 
parabola. 

A  tangent  is  a  line  touching  the  curve  at  a 
point,  and  if  produced,  does  not  cut  the  curve. 
Thus,  AC  \s  Q.  tangent,  at  the  point  B. 

7.  An  ordinate  to  a  diameter  is  a  straight  line 
drawn  from  any  point  in  the  curve  to  meet  the 
diameter,  and  is  parallel  to  a  tangent  passing 
through  the  vertex  of  that  diameter.  Thus,  BD 
is  a  diameter,  and  FD  an  ordinate  from  the  point 
F,   FD  is  parallel  to  the  tangent  AB,  drawn  through  the  vertex  B. 

It  will  be  proved  in  proposition  15,  that  FD=J)0;  and  hence, 
FG  is  called  a  doiille  ordinate. 

8.  An  abscissa  is  the  part  of  a  diameter  between  the  vertex  and 
an  ordinate.  Thus,  BJ)  is  an  abscissa,  and  DF  is  its  correspond- 
ing ordinate. 

9.  The  parameter  of  any  diameter  is  the  double  ordinate  which 
passes  through  the  focus.  Thus,  Ilf,  which  is  parallel  to  AB,  and 
passes  through  the  focus  F,  is  the  parameter  of  the  particulai* 
diameter  BD. 

10.  The  parameter  to  the^  principal  diameter  is  called  the  prin- 
cipal parameter,  or  latus-redum. 

In  a  general  sense,  the  parameder,  or  latus-recium,  means  the  con- 
stant quantity  that  enters  into  the  equation  of  a  curve.  In  a  parabola 
it  is  a  third  proportional  to  any  abscissa,  and  its  ordinate. 


THE    PARABOLA. 


249 


1 1 .  A  normud  is  a  line  drawn  perpendicu- 
lar to  a  tangent  from  its  point  of  contact,  and 
is  terminated  by  the  axis. 

12.  A  subnormal  is  the  part  of  the  axis 
intercepted  between  the  normal  and  the  cor- 
responding ordinate. 

Thus,  PC  is  a  normal,  and  DO  is  the  corresponding  subnormal, 
or  line  under  the  normal.  Similarly,  ITD  is  a  line  under  the  tangent, 
and  is  called  a  svhtangent. 


PROPOSITION    1.     THEOREM. 

The  latzis-rectum  is  four  times  the  distance  from  the  focus  to  the 
vertex. 

Let  FVJI  be  a  parabola,  F  the  focus,  and  V 
the  principal  vertex.  PJI,  at  right  angles  to  DF, 
through  the  point  F,  is  the  latus-rectum. 

We  are  to  prove  that  PH=AFV. 

Because  PH  is  parallel  to  CGy  and  C7P,  GE, 
parallel  to  DFy  the  two  figures,  OF  and  FOy  are 
parallelograms. 

Therefore,        .  CP=DF,  and  QH^BF 

Or,  .         CP'\'OH=^DF        (1) 

But  by  the  definition  of  the  curve, 

JDF=%VF,  CP^PFy  and  GH=HF 

Substitute  these  values  in  equation  (1),  and  we  have 

PF+FH==PH=^iFV.  Q.  E,  D, 

Cor,  As  CP=PFy  and  the  angles  at  F^  i),  and  C,  right  angles, 
PFDC  is  a  square. 


PROPOSITION    2.     THEOREM. 

Any  point  toithin  a  parabola  is  nearer  to  the  focus  than  to  the 
directrix;  and  any  point  without  a  parabola  is  at  a  greater  distance 
from  the  focus  than  from  the  directrix. 


250 


CONIC    SECTIONS 


Let  A  be  any  point  within  the  curve,  and 
from  it  draw  AB  perpendicular  to  the  directrix. 

As  A  is  within  the  curve,  AB  must  neces- 
sarily cut  the  curve  in  some  point.  Let  P  be 
that  point,  and  join  PF  and  AF. 

By  the  definition  of  the  curve,  PB=PF, 
To  each  of  these  add  PA,  and  AB=AP-{-PF. 
But  AP-\-PF  are,  together,  greater  than-4i?^,  because  a  straight 
line  is  the  shortest  distance  between  two  points ;  therefore,  AB  is 
greater  than  AF. 

Again,  let  A'  he  &  point  without  the  curve — ^it  is  nearer  to  the 
directrix  than  to  the  focus. 

Draw  A'F;  and  as  A'  is  without  the  curve,  this  line  must 
necessarily  meet  the  curve  in  some  point,  as  P.  Draw  PB  and 
A'B'  perpendicular  to  the  directrix,  and  join  A'B, 

A'P-\-PB=A'F 

But,         .  A'P+PB  ':>A'B;  that  is,  A'F^A'B 

But  A'Bj  being  the  hypotenuse  of  the  right  angled  triangle 
A'B'B,  it  is  greater  than  A'B\  But  A'F  is  greater  than  A'B; 
much  more  then  is  A'F  greater  than  A'B';  therefore,  any 
point,  (fee.     Q.  E.  D. 


PROPOSITION    3.     THEOREM. 

The  line  which  bisects  the  angle  which  is  formed  by  the  two  lines 
drawn  from  any  'point  in  the  curve ,  one  to  the  focus,  the  other  perpen- 
dicular to  the  directrix,  is  a  tangent  to  the  curve  at  that  point. 

Let  P  be  any  point  in  the  curve. 
Draw  PF  to  the  focus,  and  PB  per- 
pendicular to  the  directrix.  Let  PT 
be  so  drawn  as  to  bisect  the  angle 
BPF.  Then  PT  will  touch  the  para- 
bola at  the  point  P,  and  be  tangent  to 
the  curve. 

Join  BF,  and  PBF  is  an  isosceles  triangle  ;  therefore,  the  angle 
PBI=  the  angle  PFL  The  angle  BPI=  the  angle  FPl,  by 
hypothesis ;  hence,  the  two  triangles  BPI  and  PIF,  being  equi- 


JB' 

JL 

\         y 

^w 

)/, 

// 

y 

4n/ 

' 

z,_ 

A  yi- 

T  C 

i\f 

ID 

THEFARABOLA.  251 

angular,  and  having  PI  common,  are  in  all  respects  equal,  and 
PI  is  perpendicular  to  BF,  and  BI=FL 

It  now  remains  to  be  shown  that  any  other  point  than  P,  in  the 
line  APT,  is  without  the  curve. 

Take  any  other  point  in  the  line  TP,  as  A,  and  draw  the  dotted 
lines -4i^  and  ^jB.     They  are  equal.     (Th.  15,  b.  1,  scholium.) 

But  AB  being  the  hypotenuse  of  the  right  angled  triangle  AB'B 
it  is  greater  than  AB';  that  is,  AF  is  greater  than  AB' ;  conse- 
quently A  is  without  the  curve,  as  proved  by  the  last  proposition. 

In  the  same  manner  it  may  be  proved  that  any  other  point  in 
the  line  -4^  is  without  the  curve,  except  the  point  P.  AT  is, 
therefore,  a  tangent  to  the  curve  at  the  point  P.     Q.  E,  D. 

Cor.  1 .  A  line  of  light,  parallel  to  the  axis,  striking  the  point  of 
the  parabola  at  P,  will  be  reflected  to  F;  because  the  angle  of 
incidence  is  equal  to  the  angle  of  reflection ;  and  the  same  will  be 
true  at  every  point  of  the  curve  ;  hence,  if  a  reflecting  mirror  have 
a  parabolic  surface,  all  the  rays  of  light  that  meet  it  parallel  with 
the  axis,  will  be  reflected  to  the  focus  ;  and  for  this  reason  many 
attempts  have  been  made  to  form  perfect  parabolic  mirrors  for 
reflecting  telescopes. 

If  a  light  be  placed  at  the  focus  of  such  a  mirror,  it  will  reflect 
all  its  rays  in  one  direction  ;  hence,  in  certain  situations,  parabolic 
mirrors  have  been  made  for  lighthouses,  for  the  purpose  of  throw- 
ing all  the  light  seaward. 

Cor.  2.  The  angle  BPF  continually  increases,  as  the  pencil  P 
moves  toward  F,  and  at  V  it  becomes  equal  to  two  right  angles  ; 
and  the  tangent  at  V  is  perpendicular  to  the  axis,  which  is  called 
the  vertical  tangent.  . 

Cor,  3.  Since  an  ordinate  to  any  diameter  is  parallel  to  the 
tangent  at  the  vertex,  an  ordinate  to  the  axis  is  perpendicular  to 
the  axis. 

PROPOSITION    4.     THEOREM. 

If  a  tangent  he  dravmfrom  any  point  m  the  curve  to  the  axis  pro- 
duced, the  extremities  of  the  tangent  are  equally  distant  from  the  focus. 

Let  PT  (see  figure  to  the  last  proposition)  be  a  tangent,  meet- 
ing the  curve  at  jP,  and  the  axis  at  T.     Then  we  are  to  prove  that 

PF==FT 


253 


CONIC    SECTIONS. 


PB  is  parallel  to  FT;  therefore,  the  angle  BPT=  the   angle 
PTF.     But  BPT=TPF.     (Prop.  3.) 

Hence,  the   angle  PTF=  the  angle  TPF;  consequently,  the 
triangle  TFP  is  isosceles,  and  PF=TF,     Q.  K  J). 

PROPOSITION    5.     THEOREM. 

The  suhtangent  to  the  axis  is  bisected  hy  (lie  vertex. 
From  the  point  P  (see  last  figure)  draw  Pi>,  an  ordinate  to  the 
axis.     i>7^  is  a  subtangent,  and  it  is  bisected  at  F.     As  PD  is 
parallel  to  BC^  and  PB  parallel  to  (72),  PBCD\s  a  parallelogram. 


Therefore, 

.    PB=CD 

But, 

.    PB=PF,  by  the  definition  of  the  curve 

And,        . 

.    PF=FT.     (Prop.  5.) 

Therefore, 

.         .     CD=^FT 

That  is,   . 

J)V-\-VC=-TV+VF 

But, 

VC=VF 

By  subtraction, 

J)V=TV               Q.E.D. 

C(yr.  Hence,  to  draw  a  tangent  to  any  point  P,  draw  the  ordinate 
PD,  and  take  VT=  VD,  and  join  TP;  it  will  be  a  tangent  at  P. 


PROPOSITION    6.     THEOREM. 

i/^,  from  any  point  in  a  parabola,  a  tangent  and  a  normal  be  drawn^ 
both  terminated  in  the  axis,  these  two  lines  wUl  be  chords  of  a  circle, 
of  which  the  focus  is  the  center,  and  the  distance  to  the  point  P,  the 
radius. 

Let  P  be  the  point,  F  the  focus,  and 
TVC  the  axis.  Draw  PD  perpen- 
dicular to  the  axis,  and  take  TV=  VD 
(cor.  to  last  prop.)  and  join  TP,  which 
is  the  tangent  from  P.  From  P  draw 
PC,  at  right  angles  to  TP;  then  PC, 
is  the  normal.     (Def.  11.) 

Draw  PF.  By  proposition  4,  PF—FT.  Now,  if  FP  be  made 
radius,  and  a  semicircle  described,  the  points  T,  P,  and  C,  will  be 
in  the  circumference,  and  TC  will  be  the  diameter. 


THE    PARABOLA.  253 

Hence    T PC  is  &  right  angle,    and   FP=FC,  and  TF    and 
PC,  are  chords  to  this  circle ;  therefore,  if  from  any  point  &c. 

Q.  K  J). 


PROPOSITION     7.    THEOREM. 

The  svhnormal  is  equal  to  half  the  latus  rectum. 
Take  the  figure  to  the  last  proposition.     By  the  definition  of  the 
curve.  FP=D  F-f  VF=FD-\-2  VF 

Or,        .  ^VF^FP--FD  (1) 

CD=FC—FD  (2) 

By  subtracting  (2)  from  (1),  and  observing  that  FP=iFCy  we 
have,  ^VF^CD=0 

Or,        .        .      CD=2VF 

But  CD  is  the  subnormal,  and  2  VF  is  half  the  lotus  rectum ; 
therefore,  the  subnormal  <kc.  Q.  F,  D. 

PROPOSITION     8.     THEOREM. 

If  a  "perpendicular  he  drawn  from  the  focus  to  any  tangent,  the 
point  of  intersection  will  he  in  the  vertical  tangent. 

From  the  focus  F  (see  last  figure),  draw  FB  perpendicular  to 
PTy  and  as  the  triangle  PFT  is  isosceles  (Prop.  4),  and  PF  and 
FT  the  equal  sides  ;  the  line  from  the  vertex  F,  perpendicular  to 
the  base,  bisects  the  base ;  therefore,  TB=BP, 

As  VB  and  PD  are  both  perpendicular  to  the  axis,  they  are 
tlierefore  parallel. 

Hence,  .        .        TV:  VD=TB  :  BP    (th.  17,  b.  2). 
But,       .        .        .         TV=VJ) 

Therefore,      .         .  TB=BP 

That  is,  a  line  from  F  perpendicular,  to  PT,  and  a  line  from  V 
perpendicular  to  the  axis,  both  cut  the  tangent  PT  into  two  equal 
parts,  and  therefore,  meet  in  the  same  point,  B. 

Hence :  If  a  perpendicular,  &c.  Q.  E.  D. 


254  '         CONIC     SECTIONS. 

Cor.  1 .  The  two  triangles  VBF  and  PBF^  are  similar,  for  they 
are  both  right  angled  triangles,  and  the  angle  PF£=the  angle 
VFJB. 

Hence,  .        .         VF :  FB^FB  :  PF 

That  is,  ih£  perpendicular  from  the  focus  to  any  tangent,  is  a  mean 
proportional  between  the  distances  of  the  focus  from  the  vertex,  and 
from  the  point  of  contact. 

Scholium.    From  the  preceding  proportion,  we  have 

VF'PF=:FB' 

But  VF,  remains  constant  for  the  same  parabola ;  -  therefore,  the 
distance  from  the  focus  to  the  point  of  contact  varies,  as  the  square 
of  the  perpendicular  drawn  from  the  focus  upon  the  tangent. 

PROPOSITION     9.     PROBLEM. 

Find  the  equation  of  the  curve,  or  the  mathematical  relation  between 
any  abscissa  on  the  axis,  and  its  corresponding  ordinate. 

Let  F  be  taken  as  the  zero  point. 
Put  VD=x,  PI>=y,  and  let  2p  repre- 
sent the  parameter.  As  TPC,  is  a 
right  angled  triangle,  right  angled  at 
P,  PD  is  a  mean  proportional  between 
TJ)  and  DC.     (Scho.  to  th.  17,  b.  3). 

But, TD=2x        (Prop.  5). 

And, J)C=:p  (Prop.  7). 

Therefore  by  multiplication,  TD'DC=f=2px 

By  taking  the  square  root,  y=dzj2px,  the  double  sign  shows 
two  equal  values  to  y,  the  one  above,  the  other  below  the  axis ; 
hence,  the  curve  is  symmetrical  in  respect  to  its  focus  and  axis. 

PROPOSITION    10.     THEOREM. 

The  sqitares  of  ordinates  to  the  axis  are  to  one  another,  as  their 
corresponding  abscissas. 

By  the  last  proposition,  any  ordinate  represented  by  y,  and  its 


THE    PARABOLA. 


255 


corresponding  abscissa  represented  by  «,  are  connected  together  by 
the  following  equation. 

y'^zlpx         (1) 

Any  other  ordinate  represented  by  y',  and  its  corresponding  ab- 
scissa represented  by  x\  have  a  like  connection. 

That  is,         .         .         y'^'^^px'        (2) 

Dividing  (2)  by  (1),  omitting  the  common  factor  2p,  and  we 
hare 


Or, 


r'2 :  y'^x'  :  x 


Q.  K  D. 


PROPOSITION     11.    THEOREM. 

As  the  parameter  of  the  axis  is  to  the  sum  of  any  two  ordinaies,  so 
is  the  difference  of  those  ordinates  to  the  difference  of  their  abscissas. 

Let  CV^  be  a  portion  of  a  parabola,  V 
the  vertex,  VD  the  axis,  VB  and  VD  ab- 
scissas, and  FB  and  UD  their  correspond- 
ing ordinates. 

Put      VB=:x,     VD=x',    PB^y, 

And    ED=y' 

Then,  AR=x'—x,     RE=y'-{-y,     and  CR=y'—y 

From  Proposition  10. 


y'^=i 


:^pX 


By  subtraction,  y'^ — y^=2p(x' — x) 

Or,  .        .      (y'+y)(y'—y)=2p(x'-^x) 
Or,   .        .         .      2p 
Or,   .         .         •        2p 


I       Q.  K  D. 
9:RU=GB:AR     J 


25G 


CONIC    SECTIONS, 


Chr.     Take  the  product  of  the  extremes  and  means  of  this  last 
proportion  and  we  have 

(2p)x'=y^  (Prop.  10). 


But,     . 
Bj  division. 


AB     CR'EE 


X-  y 

AR_CR'RE 

.  VD  :  AR=DU^  :  CR'RU 
That  is,  any  abscissa  of  the  axis,  is  to  any  other  diamater,  so  is 
the  square  of  the  ordinate  to  the  rectangle  of  the  segments  of  the 
double  ordinate. 


Or, 
Or, 


PROPOSITION     12,    THEOREM. 

J^  a  tangent  be  draton/rom  any  point  of  a  parabola,  and  from  any 
point  in  the  tangent  a  line  be  drawn  parallel  to  the  axis,  and  termi- 
nated in  the  double  ordinate,  this  line  vMl  be  cut  by  the  curve  in  the 
same  proportion  as  the  line  cuts  the  double  ordinate. 

Let  CT  be  a  tangent  for  the  point  C,  V  the 
vertex,  VD  the  axis,  and  CU  the  double  ordi- 
nate CI>=y     VD=x 

Take  any  point  /,  in  the  tangent,  and  draw 
IR  parallel  to  VD,  cutting  the  curve  at  A. 
Then  we  are  to  show 

That    .        .    IA'.AR=CR\  RE 

Produce  i>  F  to  T,  and  observe,  that 

DV=VT, 
Or,       ...        . 
By  similar  As, 


By  eq.  of  the  curve 
By  equality,  . 
Proposition  11, 


DT=^SlDV 
CR  ',  RI=^CD  :  DT 

=y  :  2a; 
2p  :  9,y=y  :  2a; 
CR  :  RI=^2p  :  {2y)CE 
2»  :  RE=^  CR  :  AR 


(Prop.  5). 


Prod,  term,  by  term,  2^?-  CR  ;  RI-RE^^p'  CR  :  CE-AR 


THE    PARABOLA.  257 

In  this  last  proportion  the  antecedents  are  equal ;  therefore,  the 
tonsequents  are  equal. 

Hence,  .  RI»RE=CE'AR 

Or,       .        .        BI\AR^CE:RE 
By  division,  (RI-^AR)  :  AR=(  CE—RE)  :  RE 
That  is,        .        IA:AR=CR:RE  Q.  E.  D. 

Cor.  The  same  is  true,  if  a  line  be  drawn  from  any  other  point 
of  the  tangent. 

Therefore,    .       EP  :  PO=CG  :  GE 

PROPOSITION    13.    THEOREM. 

^  am  points  be  taken  on  a  tangent,  and  from  thence  lines  be  drawn 
'parallel  to  the  axis  to  meet  the  curve,  the  length  of  such  lines  will  be  to 
each  other  as  the  squares  of  the  distances  of  the  points  from  the  point 
of  contact  measured  on  the  tangent. 

Let  CJI  be  a  tangent  to  a  parabola,  and  /  and  JT"  any  points 
taken  upon  it.  Let  D  F  be  the  axis  produced  to  T.  Draw  IR 
parallel  to  VD,  meeting  the  curve  at  A;  and  also,  draw  EG  par- 
allel to  VE,  meeting  the  curve  at  P. 

We  are  now  to  prove,  that 

lA  :HP=Cr  :  CH^ 

By  the  last  proposition,  we  have 

lA  :AR=CR:RE 

Multiplying  the  last  couplet  by  CR,  and  substituting  the  value 
of  CR*RE  taken  from  corollary  to  Proposition  11,  and 

AR'CE^ 


lA  :AR=CR': 


VD 


Dividing  the  second  and  fourth  terms  by  AR,  and  afterward 
multiplying  the  same  terms  by  VD,  observing  that  VD=  VT,  then 
we  have 

IA\  VT=zCR^:  CD^ 
17 


258  CONIC    SECTIONS 

But  by  similar  triangles. 

Therefore,  by  equality, 

lA  :  TV=:Cr  :  CT^ 
In  the  same  manner,  we  may  prove  that 

ffP:  TV=CH^:  CT^ 
Dividing  one  of  these  proportions  by  the  other,  terra  by  term. 

And,        •        .  _.i=-^^.l 

Or,  .        .        .        lA:  HP=Cn :  CH^  Q.  E.  D. 

Application,  Conceive  CH  to  be  the  direction  of  a  projectile, 
and  undisturbed  by  the  resistance  of  the  air,  or  the  force  of  gravity, 
it  would  move  along  the  line  CH^  passing  over  equal  distances  in 
equal  times.  Now  let  gravity  act  in  the  direction  of  IR,  and  as 
bodies  fall  in  proportion  to  the  squares  of  the  times  of  descent, 
therefore,  lAy  TV^  HPy  &c.,  must  be  to  each  other,  as  the  squares 
CP,  CT^,  CH^,  &c;  that  is  the  real  path  of  a  projectile  un- 
disturbed by  atmospheric  resistance  must  have  the  same  property, 
as  just  demonstrated  in  this  proposition.  In  other  words,  the  path 
of  a  projectile  is  some  parahola,  more  or  less  curved  according  to 
the  direction  and  intensity  of  the  projectile  force. 

PROPOSITION    14.     THEOREM. 

Tlic  abscissas  of  any  diameter  are  to  each  other  as  the  sqtiares  of 
tlieir  corresponding  ordinates. 

By  the  definition  of  a  diameter,  it  must  be 
the  axis,  or  parallel  to  the  axis ;  and  ordinates 
to  any  diameter  must  be  parallel  to  the  tangent 
drawn  through  the  vertex  of  that  diameter. 
Hence,  if  CS  is  a  diameter,  and  CP  a  tan- 
gent, and  ly  Ty  and  0,  any  points  on  the  tan- 
gent, and  from  thence  Hues  drawn  parallel  to  the  axis  to  meet  the 
curve,  and  from  thence  lines  parallel  to  the  tangent  to  meet  the 
diameter,  the  figures  so  formed  will  be  parallelograms,  and  their 
opposite  sides  equal. 


THE    PARABOLA.  259 

By  the  last  proposition,  lE^  TA,  <kc.,  are  to  each  other  as  Cl^, 
CT\  &c.;  that  is,  (7§,  QR,  <fec.,  are  to  each  other  as  QE'^,  BA^, 
&c.;  or  the  abscissas  are  as  the  squares  of  their  corresponding 
ordinates.     Q.  E.  D. 

Remark.  This  is  the  same  property  as  was  proved  in  relation 
to  the  axis  and  its  ordinates  in  proposition  10. 


PROPOSITION    15.     THEOREM. 

7j^  a  line  he  dravm  parallel  to  any  tangent,  and  cut  the  curve  in  two 
jpointSf  and  from  these  points  ordinates  be  dravm  to  the  axis,  and 
another  from  the  point  of  contact  of  the  tangent,  then  the  three  ordinates 
will  be  in  arithmetical  progression. 

Let  CT  be  a  tangent,  and  HE  paral- 
lel to  it.  Draw  the  ordinates  EG,  CD, 
and  m. 

Then,     .     EG+m=:2CD 

From  the  similar  triangles,  REE, 
CDT,  we  have 

EE :  KE=  CD  :  DT=^AD 

By  prop,  11,        .  Zp  :  EZ=ffK :    EE 

Therefore,  by  (th.  6,  b^)  2p  :  EL=  CD  :  2AD 

By  eq.  of  the  curve,       2p  :  2CD=CD  :  2AD 

By  comparing  the  two  preceding  proportions,  we  find  that  EL 
must  equal  2  CD.     But  by  inspecting  the  figure,  we  perceive  that 

EL=LIi-IE=m+EG 

That  is,        .        .  JII-\-EG=2CD         Q.  E.  D. 

Scholium.  As  CD  is  the  arithmetical  mean  between  GE  and 
HI,  if  we  draw  CM  parallel  to  AI,  and  draw  JO^  parallel  to  CD, 
it  will  equal  CD;  hence,  JI/2V  being  midway  in  value  between  EG 
and  HI,  and  parallel  to  them,  it  must  meet  the  lines  HE  and  GI 
in  their  midway  points.  That  is,  the  diameter  CM  cuts  its  ordinate 
HE  in  two  equal  parts;  and  as  HE  is  any  ordinate,  therefore,  the 
diameter  cttts  all  its  ordinates  into  two  equal  parts. 


360  *         CONIC    SECTIONS 


PROPOSITION    1«.     THEOREM. 

A  parabola  is  a  conic  section^  the  cone  being  cut  by  a  plane  parallel 
to  its  side. 

Let  the  cone  be  cut,  or  conceived  to  be  cut,  by 
the  plane  VMN  passing  through  its  axis,  and 
then  conceire  this  plane  cut  by  the  plane  DAI^ 
perpendicular  to  the  first  plane,  and  so  inclined 
that  AH  shall  be  parallel  to  VM, 

Draw  MN  and  KL  perpendicular  to  the  axis 
of  the  cone,  and  make  them  diameters  of  parallel  circles,  whose 
planes  are  at  right  angles  to  the  plane  VMN, 

From  the  points  F  and  Hf  where  AH  meets  KL  and  J/zV, 
draw  FO  and  HI  at  right  angles  to  AH;  and  because  the  plane 
DAI  is  at  right  angles  to  the  plane  VMN",  FO-  is  at  right  angles 
to  KL,  and  HI  is  at  right  angles  to  MX. 

Now,  from  the  similar  triangles,  AFL,  AHK,  we  have 

AF :  AH^FL  :  HN 

By  reason  of  the  parallels,  KF=MH;  therefore,  by  multiplying 
the  last  couplet  we  have 

AF \  AH=FL' KF'.HN^MH 

But,  by  reason  of  the  semicircles  MIN^  KQL, 

KF'FL=FCP,  ajidMHHN=Hr  (th.  17,  b.  3.) 

Consequently,         .     AF :  AH=FO^  :  Hr 

This  is  the  same  property  as  was  demonstrated  in  proposition  10; 
therefore,  the  nature  of  the  curve  is  the  same.     Q.  E.  D. 

FG^    HI^         FG^         HI 
Cor.  Hence,    —r7=  =—rF7  and  -r^^,  or  —rrr-  is  a  third  propor- 
AF     AH         AF         AH  ^    ^ 

tional,  and  a  constant  quantity,  which  we  have   called  2p,  the 
parameter  by  definition  10. 

Remark.  We  might  have  commenced  the  subject  of  the  para- 
bola by  assuming  it  a  conic  section  of  this  kind,  and  then  sought 
out  its  other  properties. 


THE    PARABOLA. 


261 


PROPOSITION    17.     THEOREM. 

Every  segmmt  of  a  parabola  at  right  angles  with  its  axis,  is  two* 
thirds  of  its  circumscribing  rectangle. 

Let  P  be  any  point  in  the  curve,  and  PT 
a  tangent.  Draw  PB  and  BT,  Take 
any  very  small  portion  of  the  tangent,  as  PI- — 
so  small  as  to  consider  it  as  coinciding  with  the 
curve,  without  sensible  errors.  Draw  10,  Ig, 
making  the  two  rectangles  BR,  HD, 

Let  us  now  investigate  the  relation  between 
these  two  rectangles. 

PD=iy,     VD=x;     then,    PB=x,    and 


BB=x(PB),    and    HD=y{RI) 


2x 


As  customary,  put 
J)T=2x.     (Prop.  5.) 
The  rectangle 
By  similar  triangles 

PR  :  RI=y 

Multiply  the/r5/  and  third  terms  of  this  proportion  by  x,  and 
the  second  and  fourth  by  y.    We  then  have 

x(PR)  :  y{RI)=xy  :  2xy 

=  1    :  2 

The  whole  rectangle  BYDP  is  divided  into  two  spaces  by  the 
curve — the  one  within  the  curve,  the  other  external  to  it.  And  we 
perceive  by  the  above  proportion  that  the  small  rectangle,  BR, 
external  to  the  curve,  is  to  its  corresponding  rectangle,  HD,  within 
the  curve,  as  1  to  2. 

By  taking  any  other  small  portion  of  the  curve,  as  well  as  PI, 
and  drawing  its  external  and  internal  rectangle,  we  can  prove  in  the 
same  manner  that  they  will  be  to  each  other  as  1  to  2;  and  thus 
we  can  fill  up  the  whole  external  and  internal  spaces,  and  they  will 
be  to  each  other  as  1  to  2.  Hence,  the  space  within  the  curve  is 
two-thirds  of  the  whole  rectangle  BD,  and  the  same  is  true  of  the 
spaces  on  the  other  side  of  the  axis.  Therefore,  every  segment, 
&c.     C-  •^'.  ^' 


CONIC    SECTIONS 


PROPOSITION    18.     THEOREM. 

If  a  parabola  revolve  on  its  axis,  the  solid  generated  is  equal  to 
one  half  of  its  circumscribing  cylinder. 

Take  the  figure  to  the  last  proposition,  and  conceive  the  parabola 
to  revolve  on  the  axis  VD,  and  find  the  relation  between  the  two 
solids  generated  by  the  two  parallelograms  BR  and  HD.  Tlie 
parallelogram  HD  will  generate  a  cylinder^  wliose  diameter  is  2y, 
and  length  RL 

The  parallelogram  BR  will  generate  a  circular  band,  whose  length 
is  ar,  and  thickness  PR. 

The  solidity  of  the  cylinder  =7ty\RI) 

The  soHdity  of  the  band       =^{rty''—n{y—PRy)x 

These  two  quantities  are  in  the  proportion  of 

(2y(FR)^PR')x 

By  rejecting  the  very  small  quantity   (PRy  as  bemg  very  in- 
considerable in  connection  with  the  other  term,  we  have 
Sol.  of  cylinder  :  sol.  of  band  =y\RI)  :  2xy{PR) 
But,  as  in  the  preceding  proposition, 

PR:  RI=y  :  2x 
Or,  .  .  .  2x{PR)=y(RI) 
Or,         .        .         .2xy(PR)=y^RI) 

This  equation  shows  that  the  last  terms  in  the  preceding  propor- 
tion are  equal ;  therefore, 

sol.  of  cyhnder  :  sol.  of  band  =1:1 

Or  the  solidities  of  the  cylinder  and  band  are  equal ;  and  the 
same  is  true  of  every  pair  of  corresponding  solids ;  and  the  sum 
of  the  paraboloid  is  all  the  minute  cylinders  which  make  up  the 
sohd  generated  by  the  revolution  of  the  parabola,  (called  a 
parabaloid);  and  the  sum  of  all  the  minute  bands  makes  up  the 
soHd  exterior  to  the  parabaloid.  Hence,  the  parabaloid  is  equal  to 
half  its  circumscribing  cylinder.     §.  E.  D, 


THE  HYPERBOLA.  263 


THE  HYPERBOLA. 

DEFINITIONS. 

1 .  An  hyperbola  is  a  plane  curve,  confined  by  two  fixed  points 
called  the  foci,  and  the  difference  of  the  distances  of  each  and 
every  point  in  the  curve  from  the  two  fixed  points,  is  constantly 
equal  to  a  given  line. 

Remark  1.  The  distance  between  the  foci,  is  also  supposed  to  be 
known  ;  and  the  given  line  must  be  less  than  the  distance  between  the 
fixed  points  ;  that  is,  less  than  the  distance  between  the  foci. 

Remark  2.  The  ellipse  is  a  curve,  confined  by  two  fixed  points 
called  the  focii  and  the  sum  of  two  lines  drawn  from  any  point  in  the 
curve,  is  constantly  equal  to  a  given  line.  In  the  hyperbola,  the  differ- 
ence of  two  lines  drawn  from  any  point  in  the  curve,  to  the  fixed  points, 
IS  equal  to  the  given  line.  The  ellipse  is  but  a  single  curve,  and  the 
foci  are  within  it ;  but  it  will  be  shown  in  the  course  of  our  investiga- 
tion, that  the  hyperbola  consists  of  two  equal  and  apposite  branches,  and 
the  least  distance  between  them  is  the  given  line. 

2.  The  line  joining  the  foci,  and  produced,  if  necessary,  is 
called  the  axis  of  the  hyperbola. 

3.  The  middle  point  of  the  straight  line  which  joins  the  foci,  is 
called  the  center  of  the  hyperbola. 

4.  The  excentricity,  is  the  distance  from  the  center  to  either  focus. 
6.  A  diameter  is  any  straight  line  passing  through  the  center  and 

terminated  by  two  opposite  hyperbolas. 

6.  The  extremities  of  a  diameter  are  called  its  vertices. 

7.  A  tangent  is  a  straight  line  which  meets  the  curve  only  in  one 
point,  and  being  produced,  does  not  cut  the  curve. 

8.  An  ordinate  to  a  diameter,  is  a  straight  line  drawn  from  any 
point  of  the  curve  to  meet  the  diameter  produced,  and  is  parallel 
to  XhQ  tangent  at  the  vertex  of  the  diameter. 

9.  An  abscissa,  is  the  distance  between  the  tangent  point  and  its 
conesponding  ordinate,  measured  on  the  diameter  produced. 


264 


CONIC    SECTION 


10.  The  parameter  is  a  double  ordinate,  passing  through  the 
focus.  The  principal  parameter  passes  through  the  focus  at  right 
angles  to  the  axis. 

Remark.  Thus,  let  F'F  be  two  fixed  points. 
Draw  a  line  between  them,  and  bisect  it  in  0. 
Take  CA,  CA',  each  equal  to  half  the  given 
line,  and  CA  may  be  any  distance  less  than  CF; 
A' A  is  the  given  line,  and  is  called  the  major* 
axis  of  the  hyperbola.  Now  let  us  suppose  the 
curve  already  found  and  represented  by  ADF,  Take  any  point,  as 
P,  and  joinPi^  and  FF';  then  by  Definition  1,  the  difference  be- 
tween PF'  and  PF  must  be  equal  to  the  given  line  A' A,  and 
conversely  if  PF' — PF=A'A,  then  i'  is  a  point  in  the  curve. 

By  taking  any  point,  P,  in  the  curve,  and  joining  PF  and  PF\ 
a  triangle  PFF'  is  always  formed,  having  F'F  for  its  base  and  A' A 
for  the  difference  of  the  sides ;  and  these  are  all  the  conditions  ne- 
cessary to  define  the  curve. 

As  a  triangle  can  be  formed  directly  opposite  to  PF  F,  which 
shall  be  in  all  respects  exactly  equal  to  it,  the  two  triangk  s  having 
F'F  for  a  common  side  ;  the  difference  of  the  other  two  sides  of 
this  opposite  triangle  will  be  equal  to  A' A,  and  correspond  with  the 
condition  of  the  curve ;  hence,  a  curve  can  be  formed  about  the 
focus  F'  exactly  similar  and  equal  to  the  curve  about  the  focus  F. 

In  short,  F'  and  A'  have  the  same  situation 
in  respect  to  (7,  as  F  and  A  have  to  (7,  and  the 
hne  FF'  is  common  to  all  the  points  ;  therefore 
if  a  curve  can  pass  about  the  focus  F,  a  like 
curve  can  pass  about  the  focus  F'^  and  this  is 
illustrated  by  the  adjoining  figure,  representing 
a  plane  cutting  vertical  cones. 

Any  line  drawn  through  C,  and  terminated 
by  the   opposite  curves,  is  called  a  diameter; 
thus,  DD'  is  a  diameter,  and  by  a  very  simple  demonstration  we 
can  prove  that  it  is  bisected  in  C. 


*The  term  tnajor  axis  implies  that  there  is  a  minor  axis,  but  where  it  is,  we 
cannot  at  present  determine  ;  when  we  find  such  a  line,  we  will  give  it  its 
proper  name. 


THE    HYPERBOLA 


265 


PROPOSITION     1.     PROBLEM. 

To  describe  an  hyperbola. 

Take  a  ruler  F'H,  and  fasten 
one  end  at  the  point  F',  on  which 
the  ruler  may  turn  as  a  hinge.  At 
the  other  end  of  the  ruler  attach  a 
thread,  and  let  it  be  less  than  the 
ruler  by  the  given  line  A' A.  Fasten 
the  other  end  of  the  thread  at  F, 

With  a  pencil,  P,  press  the  thread  against  the  ruler  and  keep  it 
at  equal  tension  between  the  points  H  and  F.  Let  the  ruler  turn 
on  the  point  F\  keeping  the  pencil  close  to  the  ruler  and  letting 
the  thread  slide  round  the  pencil ;  the  pencil  will  thus  describe  a 
curve  on  the  paper. 

If  the  ruler  be  changed  and  made  to  revolve  about  the  other 
focus  as  a  fixed  point,  the  opposite  branch  of  the  curve  can  be 
described. 

In  all  positions  of  P,  except  when  at  A  or  A'y  PF'  and  PF  will 
be  two  sides  of  a  triangle,  and  the  difference  of  these  two  sides  is 
constantly  equal  to  the  difference  between  the  ruler  and  the  thread  ; 
but  that  difference  was  made  equal  to  the  given  line  A' A;  hence, 
by  Definition  1,  the  curve  thus  described,  must  be  an  hyperbola. 

PROPOSITION     2.     THEOREM. 

If  two  straight  lines  be  drawn  from  a  point  without  an  hyperbola 
to  the  foci,  the  excess  of  the  one  above  the  other  will  be  less  than  the 
major  axis;  but  if  the  two  straight  lines  be  dravm  from  a  point  within 
an  hyperbola  to  the  foci,  the  excess  of  one  above  the  other  will  be  greater 
than  the  major  axis. 

Explanatory  Note.  In  this 
and  all  subsequent  propositions, 
we  shall  consider  but  one  branch 
of  the  curve ;  that  about  the 
focus  F. 

The    distance    between    any 
point,  P,  on  tlie  curve,  and  the  focus  P,  will  be  represented  by  r,  and 
between  P  and  the  focus  P '  by  r'. 


266  CONIC    SECTIONS. 

Let  /  be  a  point  without  the  curve  ;  join  IFy  IF',  and  as  F  is 
within  the  curve,  the  line  IF  necessarily  cuts  the  curve  at  some 
point  P.     Let  the  line  without  the  curve  be  represented  by  h. 

Put  F'I^=.z\  and  corresponding  to  the  nature  of  the  curve,  put 
r' — r=a,  or  /=r+a. 

Add  h  to  both  members  of  this  last  equation,  and 

But  the  first  member  of  this  equation  is  the  sum  of  two  sides  of 
a  triangle,  and  of  course  greater  than  its  third  side  z  ;  therefore, 
increase  «'  by  <  to  make  it  equal  to  r'+A. 

Then,     .         .         .      2'+/=:(r+A)+a 

Or,         .         .      2'— (r+A)=a— < 

That  is,  the  difference  between  //"and  IF,  is  less  than  a,  the 
major  axis.  In  a  similar  manner,  we  may  demonstrate  that 
HF^HF  is  greater  than  a.  Q-  E,  D. 

PROPOSITION     3.     THEOREM. 

A  tangent  to  the  hyperbola  bisects  the  angle  contained  by  lines  drawn 
from  the  point  of  contact  to  the  foci. 

Let  F\  F  be  the  foci  and  P  any  point  on  the  curve,  draw  PF' 
PF  and  bisect  the  angle  F'PF  by  the  line  TT';  this  line  will  be 
a  tangent  at  P. 

If  TT'  be  a  tangent  at  P,  every  other 
point  on  this  line  will  be  without  the 
curve. 

Take  PQ=PF  and  join  OF,  TT' 
bisects  QF,  and  any  point  in  the  line 
TT'  is  at  equal  distances  from  /'and  Q- 
(th.  15  b.  1).  By  the  definition  of  the  curve  F'O^A'A  the 
given  line.  Now  take  any  other  point  than  P  in  TT'  as  E,  and 
join  EF\  EF  and  EQ,        EF=EQ. 

Therefore,  .  EF'—EF—EF'—EG.  But  ^/''—/'(7,  is  less 
than  F'  G,  because  the  difference  of  any  two  sides  of  a  triangle  is 
less  than  the  third  side  (th.  18  b.  1  ).  That  is,  EF'—EF  is  less 
than  A' A;  consequently  the  point  E  is  without  the  curve  (Prop.  2), 


THE    HYPERBOLA.  267 

and  as  E  is  any  point  on  the  line  TT'  except  P;  therefore,  the 
line,  TT\  which  bisects  the  angle  at  P,  is  a  tangent  to  the  curve  at 
that  point.  Q.  E,  D. 

Scholium.  It  should  be  observed,  that  the  variable  point  in  the 
curve,  as  P  joined  to  the  two  invariable  points  E'  and  E  form  a 
triangle,  and  that  the  tangent  of  the  curve  at  the  pomt  P,  bisects 
the  angle  of  that  triangle  at  P. 

But  when  any  angle  of  a  triangle  is  bisected,  the  bisecting  line 
cuts  the  base  into  segments  proportional  to  the  other  sides 
(th.  23  b.  2). 

Therefore,     .        .     ET  :  PE=ET  :  T'E 
Or,       ...        .     r'  :r=E'T'  :T'F 
But  as  r  must  be  greater  than  r  by  a  given  quantity  a. 
Therefore,     .         .         r-^-a  :  r=:^E'T'  :  T'E 

Or,       .        .         .         1+-  :  \=E'T'  :  T'E 

r 

Let  it  be  observed,  that  a  is  a  constant  quantity,  and  r  a  variable 
one,  which  can  increase  without  limit,  and  when  r  is  immensely/ 

great  in  respect  to  a,  the  fraction  -  is  extremely  minute^  and  the  first 

term  of  the  above  proportion,  does  not  in  any  practical  sense  differ 
from  the  second  ;  therefore,  in  that  case,  the  third  term  does  not 
essentially  differ- from  the  fourth;  that  is,  E'T'  does  not  essentially 
differ  from  FT'  when  r,  or  the  distance  of  P  from  E  is  immensely 
great.  Hence,  the  tangent  at  any  point  P,  of  the  hyperbola,  can  never 
cross  the  line  EE'  at  its  middle  point,  but  it  may  approach  within 
tho  least  imayinable  distance  to  that  point. 


268  CONIC    SECTIONS 


THE    ASYMPTOTES. 

The  direction  of  a  line  passing  through  the  center  of  opposite  hy- 
perbolas to  which  a  tangent  may  approach  within  the  least  imaginable 
distance  is  called  an  asymptote. 

PROPOSITION    4.     PROBLEM. 

To  draw  an  asymptote  to  an  hyperbola  and  find  its  angle  with  the  axis. 

Let  FF'  be  the  foci  of  an  hyperbola 
and  A' A  the  major  axis,  and  C  the 
center.  From  JP'  as  a  center  with  a 
radius  equal  A' A,  describe  a  circle. 
From  the  other  focus  F,  draw  FH  a 
tangent  to  this  circle^  and  from  the 
center  F'  and  through  the  point  of 
contact  Hj  draw  the  line  F  'H,  and  let 
it  be  indefinitely  produced.  From  C,  draw  CP  parallel  to  FH,  and 
from  F,  draw  FI  also  parallel  to  F'H;  then  the  three  lines  F'H,  CP 
and  FI,  are  all  perpendicular  to  FH,  and  therefore,  will  never  meet, 
however  far  they  may  be  produced. 

Now  suppose  F  'H  and  FI  to  make  the  slightest  possible  inclination 
toward  CP,  and  if  they  equally  incline,  it  is  evident  that  they  would 
meet  in  the  same  point  P,  and  the  less  the  inclination  from  right  angles, 
the  greater  iht  distance  to  P,  and  PHF  would  form  an  isosceles  triangle, 
having  FH  for  its  base,  and  PH,  PF  for  its  equal  sides,  and  if  PH 
and  PF  are  anything  less  than  infinity,  the  point  P  will  be  in  the 
hyperbola  ;  for,  by  our  supposition  the  infinitely  slight  inclination  at  H, 
does  not  prevent  us  from  taking  PF  'F  as  a  triangle,  and  the  difference 
of  the  sides  PF',  PF,  isF'H=^A'A. 

Hence  CP  is  a  line  to  which  the  curve  can  constantly  approach,  but 
never  meet,  or  can  meet  it  only  at  an  infinite  distance,  and  this  line  is 
called  an  asymptote. 

To  obtain  an  expression  for  its  angle  with  FF '  we  observe  that  the 
triangle  F'HF  is  right  angled  at  H,  and  FF'  and  A' A  are  always  con- 
sidered as  known  lines,  but  A'A=F'H. 

Hence,         .  FF  :  A'A=sin.  90°  :  sin.HFF',  or  aos.PCF 

In  analytical  geometry      A'A=a,  and  AF=c; 

Therefore,    .        .        .    FF'=a+2c,          F'H=a 

And,     ....     FH'=V4ac+4c2=2V^fC+c2 


THE    ASYMPTOTE 


269 


If  from  the  point  A,  we  draw  Ah  at  right  angles  to  FC,  the  two 
triangles  F'HF,  CAh,  will  be  similar,  and  give  the  proportion 
F'H:HF=CA  :  Ah 

That  is,         .       a  :  2  »J a  c-\-c^=^a  :  Ah=J{a-\-c)c 

From  the  preceding  equation,  we  perceive  that  Ah  is  a  mean  propor- 
tional between  FA  and  AF'. 

The  double  of  the  line  A  A,  drawn  at  right  angles  to  FF '  through  the 
point  C,  is  what  mathematicians  have  arbitrarily  termed  the  minor  axis. 
Hence,  they  give  this  rule  for  drawing  an  asymptote. 

Rule. — From  either  vertex  of  the  major  axis  draw  a  line  at  right  angles 
to  that  axis  equal  to  half  the  minx)r  axis,  connect  the  center  C  to  the  other 
extremity,  and  the  connecting  line  produced  is  the  asymptote. 

PROPOSITION    5.     PROBLEM. 

To  descrilfe  an  hyperbola  by  points. 

Let  F,  F'  he  the  foci  and  A' A  the 
major  axis,  and  C  the  center. 

From  F '  as  a  center  with  A' A  ra- 
dius, describe  a  portion  of  a  circle  as 
represented  in  the  figure.  From  F', 
draw  any  line  as  F'P,  cutting  the 
circle  in  H  and  join  FH.  From  F, 
draw  the  line  FP,  making  the  angle 

HFP=PHF 

It  is  obvious,  then,  that  P  must  be  in  the  curve.  In  the  same 
manner  we  find  P',  or  any  other  point.  By  joining  the  points  P  and 
C,  and  producing  it  so  that  PC=^Cp,  we  shall  have  p,  a  point  in  the 
opposite  branch  of  the  hyperbola,  and  in  the  same  manner  we  can  find 
other  points  in  the  opposite  branch. 

PROPOSITION    6.     PROBLEM. 

Find  the  equation  of  the  curve  in  relation  to  the  center  and  major  axis. 

Let  F'  F,  be  the  foci,  C  the  center,  and  A' A  the  major  axis.  Take 
any  point,  P,  on  the  curve,  and  draw  the  perpendicular  PH,}om  PF  PF'. 

Put  CA=a,  AF',  AF=c,  CF=d, 
CH=x,    PH=y,    PF=r,    PF'^r', 

Then  FH=x—d,  or  if  H  falls  be- 
tween A  and  F,  then  FH=d — x,  but  in 
either  case  the  result  will  be  the  same, 
because  (x — dy={d — xy. 


270  CONIC    SECTIONS. 

By  the  definition  of  the  curve,  we  have 

r'— r=2a  (1) 

The  A -Pi?^' gives        .       r'^=(d-\-xy+y^  (2) 

The  A  PHF  gives        .        r^=(x-~d)^+y^  (3) 

By  subtraction,        .         r'^ — r^=4dx  (4) 

Divide  (4)  by  (1)  and  r'-|-r= —  (5) 

Subtract  (1)  from  (5)  and      2r  = — —2a  (6) 

dx 
Or, **=^— «  C^) 

Combining  (7)  and  (3)  —^  ^2dx-{-a^=x^-^2dx-\-d^-{-y^ 

Or,        .         .         .      '(d'^—a^)x^=^{d?—a^)a^-\-aY     (8) 

But  the  quantity  (d^ — a^)  is  called  the  square  of  half  the  minor  axis 

by  common  consent,  and  it  is  designated  by  h"^;  a  is  half  the  major 

axis;  therefore, 

hW=a'^h^'\-aY  (9) 

Or,        .        .        .      a^ — b^x^= — a^b^    the  equation  of  the  curve. 

By  giving  different  values  to  x,  the  corresponding  values  of  y  may  be 
found.  If  we  make  x=aj  y  becomes  o,  which  shows  that  the  curve 
commences  at  the  point  A.  If  we  make  x=:a,  y  again  becomes  o, 
showing  the  opposite  point  in  the  other  branch  of  the  curve.  If  we 
make  x  less  than  a,  y  becomes  imaginary,  showing  that  there  is  no 
curve  in  a  perpendicular  direction  between  A'  and  A. 

If  in  equation  (8)  we  make  a:=<i,  PH  or  y  will  be  half  the  param- 
eter by  the  definition  of  parameter.    The  equation  then  becomes 

d'^—a''d^=:^aH''—a^-^aY 
Or,        .        .      d^—2aH^-\-a^=aY 
Or,        .        .        .  d^ — a^-=^ay 

Or -=y 

Hence,  .        .        .        .    a  :  Z>=&  :  y 

That  is,  the  parameter  is  a  third  proportional  to  the  major  and  minor 
axes. 

There  are  many  other  properties  of  the  hyperbola  not  here  demon- 
strated, but  being  of  little  or  no  practical  importance,  we  omit  them. 


LOGARITHMIC  TABLES; 


ALSO   A    TABLE   OF 


NATURAL    AND    LOGARITHMIC 


SINES,  COSINES,  AND  TANGENTS, 


TO   EVERY   MINUTE    OF   THE    QUADRANT. 


LOGARITHMS    OF 

NUMBERS 

TROM 

1    TO    10000. 

N. 

-       Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

1 

0  000000 

26 

1  414973 

51 

1  707570 

76 

1  880814 

2 

0  301030 

27 

1  431364 

52 

1  716003 

"77 

1  886491 

3 

0  477121 

28 

1  447168 

53 

1  724276 

78 

1  892095 

4 

0  602060 

29 

1  462398 

54 

1  732394 

79 

1  897627 

5 

0  698970 

30 

1  477121 

55 

1  740363 

80 

1  903090 

6 

0  778151 

31 

1  491362 

56 

1  748188 

81 

1  908485 

7 

0  845098 

32 

1  505150 

57 

1  755875 

82 

1  913814 

8 

0  903090 

33 

1  518514 

58 

1  763428 

83 

1  919078 

9 

0  954243 

34 

1  531479 

59 

1  770852 

84 

1  924279 

10 

1  000000 

35 

1  544068 

60 

1  778151 

85 

1  929419 

11 

1  041393 

36 

1  556303 

61 

1  785330 

86 

1  934498 

12 

1  079181 

37 

1  568202 

62 

1  792392 

87 

1  939519 

13 

1  113943 

38 

1  579784 

63 

1  799341 

88 

1  944483 

14 

1  146128 

39 

1  591065 

64 

1  806180 

89 

1  949390 

IB 

1  176091 

40 

1  602060 

65 

1  812913 

90 

1  954243 

16 

1  204120 

41 

1  612784 

66 

1  819544 

91 

1  959041 

17 

1  230449 

42 

1  623249 

67 

1  826075 

92 

1  963788 

18 

1  255273 

43 

1  633468 

68 

1  832509 

93 

1  968483 

19 

1  278754 

44 

1  643453 

69 

1  838849 

94 

1  973128 

20 

1  301030 

45 

1  653213 

70 

1  845098 

95 

1  977724 

21 

1  322219 

46 

1  662578 

71 

1  851258 

96 

1  982271 

22 

1  342423 

47 

1  672098 

72 

1  857333 

97 

1  986772 

23 

1  361728 

48 

1  681241 

73 

1  863323 

98 

1  991226 

24 

1  380211 

49 

1  690196 

74 

1  869232 

99 

1  995635 

25 

1  397940 

50 

1  698970 

75 

1  875061 

100 

2  000000 

N.  B.  In  the  following  table,  in  the  last  ni 

ne  columns  of  each  p 

age,  where 

the  first  or  leading  figures  change  from  9'^ 

5  to  O's,  points  or  do 

ts  are  now 

introduced  instead  of  the  O's  through  the  r 

est  of  the  line,  to  cat( 

ih.  he  eye, 

and  to  indicate  that  from  thence  the  corr 

Bsponding  natural  n 

umbers  in 

the  first  column  stands  in  the  next  lower 

Zirte,  and  its  annexe 

d  first  two 

figures  of  the  Logarithms  in  the  second  cc 

lumn. 

LOGARITHMS  OF  NUMBERS.      3 

N. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

100 

000000 

0434 

0868 

1301 

1734 

2166 

2598 

3029 

3461 

3891 

101 

4321 

4750 

5181 

5609 

6038 

6466 

6894 

7321 

7748 

8174 

10-2 

8600 

9026 

94^ 

9876 

.300 

.7^4, 

1147 

1570 

1993 

2415 

103 

012837 

3259 

3680 

4100 

4521 

4940 

5360 

5779 

6197 

6616 

104 

7033 

7451 

7868 

8284 

8700 

9116 

9532 

9947 

.361 

.775 

105 

021189 

1603 

2016 

2428 

2841 

3252 

3664 

4075 

4486 

4896 

103 

5306 

5715 

6125 

6533 

6942 

7350 

7757 

8164 

8571 

8978 

107 

9384 

9789 

.195 

.600 

1004 

1408 

1812 

2216 

2619 

3021 

108 

033424 

3826 

4227 

4628 

5029 

5430 

5830 

6230 

6629 

7028 

109 

7426 

7825 

8223 

8620 

9017 

9414 

9811 

.207 

.602 

.998 

110 

041393 

1787 

2182 

2576 

2969 

3362 

3755 

4148 

4540 

4932 

111 

5323 

5714 

6105 

6495 

6885 

7275 

7664 

8053 

8442 

8830 

112 

9218 

9606 

9993 

.380 

.766 

1153 

1538 

1924 

2309 

2694 

113 

053078 

3463 

3846 

4230 

4613 

4996 

5378 

5760 

6142 

6524 

114 

6905 

7286 

7666 

8046 

8426 

8805 

9185 

9563 

9942 

.320 

115 

060698 

1075 

1452 

1829 

2206 

2582 

2958 

3333 

3709 

4083 

116 

4458 

4832 

5206 

5580 

5953 

6326 

6699 

7071 

7443 

7815 

117 

8186 

8557 

8928 

9298 

9668 

..38 

.407 

.776 

1145 

1514 

118 

071882 

2250 

2617 

2985 

3352 

3718 

4085 

4451 

4816 

5182 

119 

5547 

5912 

6276 

6640 

7004 

7368 

7731 

8094 

8457 

8819 

120 

9181 

9543 

9904 

.266 

.626 

.987 

1347 

1707 

2067 

2426 

121 

082785 

3144 

3503 

3861 

4219 

4576 

4934 

5291 

5647 

6004 

122 

6360 

6716 

7071 

7426 

7781 

8136 

8490 

8845 

9198 

9552 

123 

9905 

.258 

.611 

.963 

1315 

1667 

2018 

2370 

2721 

3071 

124 

093422 

3772 

4122 

4471 

4820 

5169 

5518 

5866 

6216 

6562 

125 

6910 

7257 

7604 

7951 

8298 

8644 

8990 

9335 

9681 

1026 

126 

100371 

0715 

1059 

1403 

1747 

2091 

2434 

2777 

3119 

3462 

127 

3804 

4146 

4487 

4828 

5169 

5510 

5861 

6191 

6531 

6871 

128 

7210 

7549 

7888 

8227 

8565 

8903 

9241 

9579 

9916 

.253 

129 

110590 

0926 

1263 

1599 

1934 

2270 

2605 

2940 

3275 

3609 

130 

3943 

4277 

4611 

4944 

5278 

5611 

5943 

6276 

6608 

6940 

131 

7271 

7603 

7934 

8265 

8595 

8926 

9256 

9586 

9915 

0245 

132 

120574 

0903 

1231 

1560 

1888 

2216 

2544 

2871 

3198 

3525 

133 

3852 

4178 

4504 

4830 

5156 

5481 

5806 

6131 

6456 

6781 

134 

7105 

7429 

7753 

8076 

8399 

8722 

9045 

9368 

9G90 

..12 

135 

130334 

0655 

0977 

1298 

1619 

1939 

2260 

2580 

2900 

3219 

136 

3539 

3858 

4177 

4496 

4814 

5133 

5451 

5769 

6086 

6403 

137 

6721 

7037 

7354 

7671 

7987 

8303 

8618 

8934 

9249 

9564 

138 

9879 

.194 

.508 

.822 

1136 

1450 

1763 

2076 

2389 

2702 

139 

143015 

3327 

3630 

3951 

4263 

4574 

4885 

5196 

5507 

5818 

140 

6128 

6438 

6748 

7058 

7367 

7676 

7985 

8294 

8603 

8911 

141 

9219 

9527 

9835 

.142 

.449 

.756 

1063 

1370 

1676 

1982 

142 

152288 

2594 

2900 

3205 

3510 

3815 

4120 

4424 

4728 

5032 

143 

5336 

5640 

5943 

6246 

6549 

6852 

7154 

7457 

7759 

8061 

144 

8362 

8664 

8965 

9266 

9567 

9868 

.168 

.469 

.769 

1068 

145 

161368 

1667 

1967 

2266 

2564 

2863 

3161 

3460 

3758 

4055 

146 

4353 

4650 

4947 

5244 

5541 

5838 

6134 

6430 

6726 

7022 

147 

7317 

7613 

7908 

8203 

8497 

8792 

9086 

9380 

9674 

9968 

148 

170262 

0555 

0848 

1141 

1434 

1726 

2019 

2311 

2603 

2895 

149 

3186 

3478 

3769 

4060 

4351 

4641 

4932 

5222 

5512 

5802 

18 


4 

LOGARITHMS 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

160 

176091 

6381 

6670 

6959 

7248 

7536 

7826 

8113 

8401 

8689 

161 

8977 

9264 

9552 

9839 

.126 

.413 

.699 

.986 

1272 

1558 

152 

181844 

2129 

2415 

2700 

2985 

3270 

3555 

3839 

4123 

4407 

163 

4691 

4975 

5259 

5542 

5826 

6108 

6391 

(it)74 

6956 

7239 

164 

7521 

7803 

8084 

8366 

8647 

8928 

9209 

9490 

9771 

.,51 

155 

190332 

0612 

0892 

1171 

1451 

1730 

2010 

2289 

2567 

2846 

166 

3125 

3403 

3681 

3959 

4237 

4514 

4792 

5069 

5346 

5623 

157 

5899 

6176 

6453 

6729 

7005 

7281 

';556 

7832 

8107 

8382 

158 

8667 

8932 

9206 

9481 

9755 

..29 

.303 

.577 

.850 

1124 

159 

201397 

1670 

1943 

2216 

2488 

2761 

3033 

3305 

3577 

3848 

160 

4120 

4391 

4663 

4934 

5204 

5475 

5746 

6016 

6286 

6556 

161 

6826 

7096 

7365 

7634 

7904 

8173 

8441 

8710 

8979 

9247 

162 

9515 

9783 

..51 

.319 

;586 

.853 

1121 

1388 

1654 

1921 

163 

212188 

2454 

2720 

2986 

3252 

3518 

3783 

4049' 

4314 

4579 

164 

4844 

5109 

5373 

5638 

5902 

6166 

6430 

6694 

6957 

7221 

165 

7484 

7747 

8010 

8273 

8536 

8798 

9060 

9323 

9585 

9846 

166 

220108 

0370 

0631 

0892 

1153 

1414 

1675 

1936 

2196 

2456 

167 

2716 

2976 

3236 

3496 

3755 

4015 

4274 

4633 

4792 

5051 

168 

5309 

5568 

5S26 

6084 

6342 

6600 

6868 

7115 

7372 

7630 

169 

7887 

8144 

8400 

8657 

8913 

9170 

9426 

9682 

9938 

.193 

170 

230449 

0704 

0960 

1215 

1470 

1724 

1979 

2234 

2488 

2742 

171 

2996 

3250 

3604 

3757 

4011 

4264 

4517 

4770 

6023 

5276 

172 

5528 

5781 

6033 

6285 

6537 

6789 

7041 

7292 

7644 

7795 

173 

8046 

8297 

8548 

8799 

9049 

9299 

9550 

9800 

..50 

.300 

174 

240549 

0799 

1048 

1297 

1546 

1795 

2044 

2293 

2541 

2790 

175 

3038 

3285 

3534 

3782 

4030 

4277 

4525 

4772 

5019 

5266 

176 

5513 

5759 

6006 

6252 

6499 

6745 

6991 

7237 

7482 

7728 

177 

7973 

8219 

8464 

8709 

8954 

9198 

9443 

9687 

9932 

.176 

178 

250420 

0664 

0908 

1151 

1395 

1638 

1881 

2125 

2368 

2610 

179 

2853 

3096 

3338 

3580 

3822 

4064 

4306 

4648 

4790 

5031 

180 

5273 

5514 

5755 

5996 

6237 

6477 

6718 

6958 

7198 

7439 

181 

7679 

7918 

8168 

8398 

8637 

8877 

9116 

9355 

9594 

9833 

182 

260071 

0310 

0548 

0787 

1025 

1263 

1601 

1739 

1976 

2214 

183 

2451 

2688 

2925 

3162 

3399 

3636 

3873 

4109 

4346 

4582 

184 

4818 

5054 

5290 

5525 

5761 

5996 

6232 

6467 

6702 

6937 

185 

7172 

7406 

7641 

7875 

8110 

8344 

8578 

8812 

9046 

9279 

186 

9513 

9746 

9980 

.213 

.446 

.679 

.912 

1144 

1377 

1609 

187 

271842 

2074 

2306 

2538 

2770 

3001 

3233 

3464 

3690 

3927 

188 

4158 

4389 

4620 

4850 

5081 

5311 

5542 

5772 

6002 

6232 

189 

6462 

6692 

6921 

7151 

7380 

7609 

7838 

8067 

8296 

8526 

190 

8754 

8982 

9211 

9439 

9667 

9895 

.123 

.351 

.578 

.806 

191 

281033 

1261 

1488 

1715 

1942 

2169 

2396 

2622 

2840 

3075 

192 

3301 

3527 

3753 

3979 

4205 

4431 

4656 

4882 

6107 

5332 

193 

5557 

5782 

6007 

6232 

6456 

6681 

6905 

7130 

7354 

7578 

194 

7802 

8026 

8249 

8473 

8696 

8920 

9143 

9360 

958^ 

9812 

195 

290036 

0257 

0480 

0702 

0925 

1147 

1369 

1591 

1813 

2034 

196 

2256 

2478 

2699 

2920 

3141 

3363 

3584 

3804 

4025 

4246 

197 

4466 

4687 

4907 

6127 

6347 

5667 

6787 

6007 

6226 

6446 

198 

6665 

6884 

7104 

7323 

7642 

7761 

7979 

8198 

8416 

8635 

199 

8853 

9071 

9289 

9507 

9725  9943 

.161 

.378 

.595 

,8.3 

OF  NUMBERS.               5 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

200 

301030 

1247  1464 

1681 

1898 

2114 

2331 

2547 

2764 

2980 

201 

3196 

3412 

3628 

3844 

4059 

4275 

4491 

4706 

4921 

5136 

202 

5351 

5566 

5781 

5996 

6211 

6425 

6639 

6854 

70ii8 

7282 

203 

7496 

7710 

7924 

8137 

8351 

8564 

8778 

8991 

9204 

9417 

204 

9030 

9843 

..56 

.268 

.481 

.693 

.906 

1118 

1330 

1642 

205 

311754 

1966 

2177 

2389 

2600 

2812 

3023 

3234 

3445 

3666 

206 

3867 

4078 

4289 

4499 

4710 

4920 

5130 

5340 

5551 

5760 

207 

5970 

6180 

6390 

6599 

6809 

7018 

7227 

7436 

7646 

7854 

208 

8083 

8272 

8481 

8689 

8898 

9106 

9314 

9522 

9730 

9938 

209 

320146 

0354 

0662 

0769 

0977 

1184 

1391 

1598 

1805 

2012 

210 

2219 

2426 

2633 

2839 

3046 

3252 

3458 

3655 

3871 

4077 

211 

4282 

4488 

4694 

4899 

5105 

6310 

5516 

5721 

5926 

6131 

212 

6336 

6541 

6745 

6960 

7155 

7359 

7563 

7767 

7972 

8176 

213 

8380 

8583 

8787 

8991 

9194 

9398 

9601 

9805 

...8 

.211 

214 

330414 

0617 

0819 

1022 

1225 

1427 

1630 

1832 

2034 

2236 

215 

2438 

2640 

2842 

3044 

3246 

3447 

3649 

3850 

4051 

4253 

216 

4454 

4655 

4856 

5057 

5257 

5458 

5658 

5859 

6059 

6260 

217 

6460 

6660 

6860 

7060 

7260 

7459 

7659 

7858 

8058 

8257 

218 

8456 

8656 

8855 

9054 

9253 

9451 

9650 

9849 

..47 

.246 

219 

340444 

0642 

0841 

1039 

1237 

1435 

1632 

1830 

2028 

2225 

220 

2423 

2620 

2817 

3014 

3212 

3409 

3606 

3802 

3999 

4196 

221 

4392 

4589 

4785 

4981 

5178 

5374 

5570 

576() 

5952 

6157 

222 

6353 

6549 

6744 

6939 

7135 

7330 

7525 

7720 

7915 

8110 

223 

8305 

8500 

8694 

8889 

9083 

9278 

9472 

9666 

9860 

.54 

224 

350248 

0442 

0636 

0829 

1023 

1216 

1410 

1603 

1796 

1989 

225 

2183 

2375 

2568 

2761 

2954 

3147 

3339 

3532 

3724 

3916 

226 

4108 

4301 

4493 

4685 

4876 

5068 

5260 

5452 

5643 

5834 

227 

6026 

6217 

6408 

6599 

6790 

6981 

7172 

7363 

7554 

7744 

228 

7935 

8125 

8316 

8506 

8696 

8886 

9076 

9266 

9456 

9646 

229 

9836 

..25 

.215 

.404 

.593 

.783 

.972 

1161 

1350 

1639 

230 

361728 

1917 

2105 

2294 

2482 

2671 

2859 

3048 

3236 

3424 

231 

3612 

3800 

3988 

4176 

4363 

4551 

4739 

4926 

5113 

5301 

232 

5488 

5675 

5862 

6049 

6236 

6423 

6610 

6796 

(  983  .  7169 

233 

7356 

7542 

7729 

7915 

8101 

8287 

8473 

8659 

8845  :  9030 

234 

9216 

9401 

9587 

9772 

9958 

.143 

.328 

.513 

.698  !  .883 

235 

371068 

1253 

1437 

1622 

1806 

1991 

2175 

23)0 

2544  '   2  28 

236 

2912 

3096 

3280 

3464 

3647 

3831 

4015 

4198 

4382  4565 

237 

4748 

4932 

5115 

5298 

5481 

5664 

5846 

6029 

6212  or 94 

238 

6577 

6759 

6942 

7124 

7306 

7488 

7670 

7852 

80^4  8216 

239 

8398 

8580 

8761 

8943 

9124 

9306 

9487 

9668 

9849  . .30 

240 

380211 

0392 

0573 

0754 

0934 

1115 

1296 

1476 

1656  1837 

241 

2017 

2197 

2377 

2557 

2737 

2917 

3097 

3277 

3456  3636 

242 

3815 

3995 

4174 

4353 

4533 

4712 

4891 

5070 

6249  5428 

243 

5606 

5785 

5964 

6142 

6321 

6499 

6677 

6856 

'■<034  7212 

244 

7390 

7568 

7746 

7923 

8101 

8279 

8456 

8634 

8811  8989 

1 

245 

9166 

9343 

9520 

9698 

9875 

..51 

.228 

.405 

.582  .759 

246 

390335 

1112 

1288 

1464 

1641 

1817 

1993 

2169 

2345  2521 

247 

2697 

2873 

3048 

3224 

3400 

3576 

3751 

3926 

4101  4277 

248 

4452 

4627 

4802 

4977 

5152 

5326 

5501 

5676 

5850  6025 

249 

6199 

6374 

6548 

6722 

6896 

7071 

7246 

7419 

7592  .  7766 

6 

LOGARITHMS 

N. 

0 
397940 

1 

2 

3 

4 

6 

6 

7 

8 

9 
9501 

250 

8114 

8287 

8461 

8634 

8808 

8981 

9154 

9328 

251 

9674 

9847 

..20 

.192 

.365 

.538 

.711 

.883 

1056 

1228 

252 

401401 

1573 

1745 

1917 

2089 

2261 

2433 

2605 

2777 

2949 

253 

3121 

3292 

3464 

3635 

3807 

3978 

4149 

4320 

4492 

46b3 

254 

4834 

5005 

5176 

5346 

5517 

6688 

5858 

0029 

6199 

6370 

256 

6540 

6710 

6881 

7051 

7221 

7391 

7561 

7731 

7901 

8070 

256 

8240 

8410 

8579 

8749 

8918 

9087 

9257 

9426 

9595 

9764 

257 

9933 

.102 

.271 

.440 

.609 

.777 

.946 

1114 

1283 

1451 

258 

411620 

1788 

1956 

2124 

2293 

2461 

2629 

2796 

2964 

3132 

259 

3300 

3467 

3636 

3803 

3970 

4137 

4305 

4472 

4639 

4806 

260 

4973 

5140 

5307 

5474 

5641 

5808 

5974 

6141 

6308 

6474 

261 

6641 

6807 

6973 

7139 

>306 

';472 

7638 

7804 

7970 

8135 

262 

8301 

8467 

8633 

8798 

8964 

9129 

9295 

9460 

9625 

9791 

263 

9956 

.121 

.286 

.451 

.616 

.781 

.945 

1110 

1275 

1439 

264 

421604 

1788 

1933 

2097 

2261 

2426 

2590 

2754 

2918 

3082 

265 

3246 

3410 

3574 

3737 

3901 

4065 

4228 

4392 

4555 

4718 

266 

4882 

5045 

5208 

5371 

5534 

5697 

5860 

6023 

6186 

6349 

267 

6511 

6674 

6836 

6999 

7161 

7324 

7486 

7648 

7811 

7973 

268 

8135 

8297 

8459 

8621 

8783 

8944 

9106 

9268 

9429 

9591 

269 

9752 

9914 

..75 

.236 

.398 

.559 

.720 

.881 

1042 

1203 

270 

431364 

1525 

1685 

1846 

2007 

2167 

2328 

2488 

2649 

2809 

271 

2969 

3130 

3290 

3450 

3610 

3770 

3930 

4090 

4249 

4^109 

272 

4569 

4729 

4888 

5048 

5207 

5367 

5526 

5685 

5844 

6004 

273 

6163 

6322 

6481 

6640 

6800 

6957 

7116 

7275 

7433 

7592 

274 

7751 

7909 

8067 

8226 

8384 

8542 

8701 

8859 

9017 

9175 

275 

9333 

9491 

9648 

9806 

9964 

.122 

.279 

.437 

.594 

.762 

276 

440909 

1066 

1224 

1381 

1538 

1695 

1852 

2009 

2166 

1'323 

277 

2480 

2637 

2793 

2950 

3106 

3263 

3419 

3576 

3732 

3889 

278 

4045 

4201 

4357 

4613 

4669 

4825 

4981 

5137 

5293 

6449 

279 

6604 

5760 

5915 

6071 

6226 

6382 

6537 

6692 

6848 

7003 

280 

7158 

7313 

7468 

7623 

7778 

7933 

8088 

8242 

8397 

8552 

281 

8706 

8861 

9015 

9170 

9324 

9478 

9633 

9787 

9941 

..95 

282 

450249 

0403 

0557 

0711 

0865 

1018 

1172 

1326 

1479 

1633 

283 

1786 

1940 

2093 

2247 

2400 

2553 

2706 

2859 

3012 

3165 

284 

3318 

3471 

3624 

3777 

3930 

4082 

4235 

4387 

4540 

4692 

285 

4845 

4997 

5150 

5302 

5454 

5606 

5758 

5910 

6062 

6214 

286 

6366 

6518 

6670 

6821 

6973 

7125 

;276 

7428 

7579 

7731 

287 

7882 

8033 

8184 

8336 

8487 

8638 

8789 

8940 

9091 

9242 

288 

9392 

9543 

9694 

9845 

9995 

.146 

.296 

.417 

.597 

.748 

289 

460898 

1048 

1198 

1348 

1499 

1649 

1799 

1948 

2098 

2248 

290 

2398 

2548 

2697 

2847 

2997 

3146 

3296 

3445 

3594 

3744 

291 

3893 

4042 

4191 

4340 

4490 

4639 

4788 

4936 

5085 

5234 

292 

5383 

5532 

5680 

5829 

5977 

6126 

6274 

6423 

6571 

6719 

293 

6868 

7016 

7164 

7312 

7460 

7608 

7756 

7904 

8052 

8200 

294 

8347 

8495 

8643 

8790 

8938 

9085 

9233 

9380 

9627 

9676 

295 

9822 

9969 

.116 

.263 

.410 

.557 

.704 

.851 

.998 

1145 

296 

471292 

1438 

1585 

1732 

1878 

2025 

2171 

2318 

i464 

2610 

297 

2756 

2903 

3049 

3195 

3341 

3487 

3633 

3779 

3925 

4071 

298 

4216 

4362 

4508 

4653 

4799 

4944 

5090 

5235 

5381 

5526 

299 

5671 

5816 

5962 

6107 

6252 

6397 

6542 

6687 

6832 

6976 

OF  NUMBERS.              7 

N. 

0 

I 

2 

3 

4 

6 

6 

7 

8 

9 

300 

477121 

7266 

7411 

7555 

7700 

7844 

7989 

8133 

8278 

8422 

301 

8566 

8711 

8855 

8999 

9143 

9287 

9481 

9575 

9719 

9863 

302 

480007 

0151 

0294 

0438 

0582 

0725 

0869 

1012 

1156 

1299 

303 

1443 

1586 

1729 

1872 

2016 

2169 

2302 

2445 

2588 

2731 

304 

2874 

3016 

3159 

3302 

3445 

3687 

3730 

3872 

4015 

4157 

305 

4300 

4442 

4585 

4727 

4869 

5011 

5153 

5295 

5437 

5579 

306 

5721 

6863 

6005 

6147 

6289 

6430 

6572 

6714 

6855 

6997 

307 

7138 

7280 

7421 

7563 

7704 

7845 

7986 

8127 

8269 

8410 

308 

8551 

8692 

8833 

8974 

9114 

9255 

9396 

9537 

9667 

9818 

309 

9959 

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.380 

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.941 

1081 

1222 

310 

491362 

1502 

1642 

1782 

1922 

2062 

2201 

2341 

2481 

2621 

311 

2760 

2900 

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3179 

3319 

3458 

3697 

3737 

3876 

4015 

312 

4155 

4294 

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4672 

4711 

4860 

4989 

5128 

6267 

5406 

313 

6544 

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63-76 

6515 

6653 

6791 

314 

6930 

7068 

7206 

7344 

7483 

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7759 

7897 

8035 

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315 

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9137 

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316 

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317 

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1196 

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1470 

1607 

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2017 

2154 

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318 

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2837 

2973 

3109 

3246 

3382 

3518 

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319 

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3927 

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5014 

320 

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324 

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325 

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326 

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327 

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7196 

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330 

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331 

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.876 

1007 

332 

521138 

1269 

1400 

1530 

1661 

1792 

1922 

2063 

2183 

2314 

333 

2444 

2675 

2706 

2835 

2966 

3096 

3226 

3356 

3480 

3616 

334 

3746 

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4006 

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4266 

4396 

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4785 

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5961 

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7243 

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8016 

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8274 

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8660 

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338 

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339 

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0456 

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0712 

0840 

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1090 

1223 

1351 

340 

1479 

1607 

1734 

1862 

1960 

2117 

2245 

2372 

2500 

2627 

341 

2764 

2882 

3009 

3136 

3264 

3391 

3618 

3645 

3772 

3899 

342 

4026 

4163 

4280 

4407 

4534 

4661 

4787 

4914 

5041 

5167 

343 

5294 

6421 

5547 

5674 

6800 

6927 

6058 

6180 

6306 

6432 

344 

6668 

6686 

6811 

6937 

7060 

7189 

7315 

7441 

7567 

7693 

345 

7819 

7945 

8071 

8197 

8322 

8448 

8574 

8699 

8825 

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346 

9076 

9202 

9327 

9452 

9578 

9703 

9829 

9954 

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347 

540329 

0465 

0680 

0705 

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1080 

1206 

1330 

1464 

348 

1679 

1704 

1829 

1953 

2078 

2203 

2327 

2452 

2576 

2701 

349 

2825 

2960 

3074 

3199 

3323 

3447 

3571 

3696 

3820 

3944 

8 

LOGARITHMS 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

350 

544068 

4192 

4316 

4440 

4564 

4688 

4812 

4936 

5060 

5183 

351 

5307 

5431 

5555 

5678 

5805 

5925 

6049 

6172 

6296 

6419 

352 

6543 

6666 

6789 

6913 

7036 

7159 

7282 

7405 

7529 

7652 

353 

7775 

7898 

8021 

8144 

8267 

8389 

8512 

8635 

8758 

8881 

354 

9003 

9126 

9249 

9371 

9494 

9616 

9739 

9861 

9984 

.196 

355 

550228 

0351 

0473 

0595 

0717 

0840 

0962 

1084 

1206 

1328 

356 

1450 

1572 

1694 

1816 

1938 

2060 

2181 

2303 

2425 

2547 

357 

2668 

2790 

2911 

3033 

3155 

3276 

3393 

3519 

3640 

3762 

358 

3883 

4004 

4126 

4247 

4368 

4489 

4610 

4731 

4862 

4973 

359 

5094 

5215 

5346 

5457 

5578 

5699 

5820 

5940 

6061 

6182 

360 

6303 

6423 

6544 

6664 

6785 

6905 

7026 

7146 

7267 

7387 

361 

7507 

7627 

7748 

7868 

7988 

8108 

8228 

8349 

8469 

8589 

362 

8709 

8829 

8948 

9068 

9188 

9303 

9428 

9548 

9667 

9787 

363 

9907 

.26 

.146 

.265 

.385 

.504 

.624 

.743 

.863 

.982 

364 

561101 

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1340 

1469 

1578 

1698 

1817 

1936 

2055 

2173 

365 

2293 

2412 

2531 

2650 

2769 

2887 

3006 

3125 

3244 

3362 

366 

3481 

3600 

3718 

3837 

3955 

4074 

4192 

4311 

4429 

4648 

367 

4666 

4784 

4903 

5021 

5139 

5257 

5376 

5494 

5612 

6730 

368 

5848 

5966 

6084 

6202 

6320 

6437 

6555 

6673 

6791 

6909 

369 

7026 

7144 

7262 

7379 

7497 

7614 

7732 

7849 

7967 

8084 

370 

8202 

8319 

8436 

8554 

8671 

8788 

8905 

9023 

9140 

9257 

371 

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9608 

9725 

9882 

9959 

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372 

570543 

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0893 

1010 

1126 

1243 

1359 

1476 

1592 

373 

1709 

1825 

1942 

2058 

2174 

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2756 

374 

2872 

2988 

3104 

3220 

3336 

3452 

3568 

360J4  3800 

£915 

375 

4031 

4147 

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4379 

4494 

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4957 

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5534 

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5765 

5880 

6996 

6111 

6226 

377 

6341 

6457 

6572 

6687 

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6917 

7032 

7147 

7262 

7377 

378 

7492 

7607 

7722 

7836 

7951 

8066 

8181 

8296 

8410 

8626 

379 

8639 

8754 

8868 

8983 

9097 

9212 

9326 

9441 

9666 

9669 

380 

9784 

9898 

..12 

.126 

.241 

.355 

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.583 

.697 

.811 

381 

580925 

1039 

1153 

1267 

1381 

1495 

1608 

1722 

1836 

1960 

382 

2063 

2177 

2291 

2404 

2518 

2631 

2745 

J858 

2972 

3086 

383 

3199 

3312 

3426 

3539 

3652 

3765 

3879 

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4105 

4218 

384 

4331 

4444 

4557 

4670 

4783 

4896 

5009 

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5236 

5348 

385 

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5686 

5799 

5912 

6024 

6137 

0250 

6362 

6475 

386 

6587 

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6812 

6925 

7037 

7149 

7262 

7374 

7486 

7599 

387 

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7823 

7935 

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8160 

8272 

8384 

8496 

8608 

8720 

388 

8832 

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9056 

9167 

9279 

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9615 

9726 

9834 

389 

9950 

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.284 

.396 

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.619 

•.730 

.842 

.953 

390 

591065 

1176 

1287 

1399 

1510 

1621 

1732 

1843 

1955 

2066 

391 

2177 

2288 

2399 

2510 

2621 

2732 

2843 

2954 

3064 

317C 

392 

3286 

3397 

3508 

3618 

3729 

3840 

3950 

4061 

4171 

4282 

393 

4393 

4503 

4614 

4724 

4834 

4945 

5055 

5165 

5276 

6386 

394 

5496 

5606 

5717 

5827 

5937 

6047 

6157 

6267 

6377 

6487 

395 

6597 

6707 

6817 

6927 

7037 

7146 

7256 

7366 

7476 

7586 

396 

7695 

7805 

7914 

8024 

8134 

8243 

8353 

8462 

8672 

8681 

397 

8791 

8900 

9009 

9119 

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9446 

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9666 

9774 

398 

9883 

9992 

.101 

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.319 

.428 

.537 

.646 

.755 

.864 

399 

600973 

1082 

1191 

1299 

1408 

1517 

1625 

1734 

1843 

1951 

OF  NUMBERS.              9 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

400 

602060 

2169 

2277 

2386 

2494 

2603 

2711 

2819 

2928 

3036 

401 

3144 

3253 

3361 

3469 

3573 

3686 

3794 

3902 

4010 

4118 

402 

4226 

4334 

4442 

4550 

4658 

4766 

4874 

4982 

5089 

5197 

403 

5305 

5413 

5521 

5628 

5736 

6844 

5951 

6059 

6166 

6274 

404 

6381 

6489 

6596 

6704 

6811 

6919 

7026 

7133 

7241 

7348 

405 

7455 

7562 

7669 

7777 

7884 

7991 

8098 

8205 

8312 

8419 

406 

8526 

8633 

8740 

8847 

8954 

9061 

9167 

9274 

9381 

9488 

407 

9594 

9701 

9808 

9914 

..21 

.128 

.234 

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.447 

.554 

408 

610660 

0767 

0873 

0979 

1086 

1192 

1298 

1405 

1511 

1617 

409 

1723 

1829 

1936 

2042 

2148 

2264 

2360 

2466 

2572 

2678 

410 

2784 

2890 

2996 

3102 

3207 

3313 

3419 

3525 

3630 

3736 

411 

3842 

3947 

4053 

4159 

4264 

4370 

4476 

4581 

4686 

4792 

412 

4897 

5003 

5108 

5213 

5319 

5424 

6529 

5634 

6740 

5845 

413 

5950 

6055 

6160 

6265 

6370 

6476 

6581 

6686 

6790 

6896 

414 

7000 

7105 

7210 

7315 

7420 

7525 

7629 

7734 

7839 

7943 

415 

8048 

8153 

8257 

8362 

8466 

8571 

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8780 

8884 

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416 

9293 

9198 

9302 

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9511 

9615 

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417 

620136 

0140 

0344 

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0662 

0666 

0760 

0864 

0968 

1072 

418 

1176 

1280 

1384 

1488 

1592 

1695 

1799 

1903 

2007 

2110 

419 

2214 

2318 

2421 

2525 

2628 

2732 

2835 

2939 

3042 

3146 

420 

3249 

3353 

3456 

3559 

3663 

3766 

3869 

3973 

4076 

4179 

421 

4282 

4385 

4488 

4691 

4695 

4798 

4901 

5004 

6107 

5210 

422 

5312 

5415 

5518 

5621 

5724 

6827 

5929 

6032 

6135 

6238 

423 

6340 

6443 

6546 

6648 

6751 

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6956 

7068 

7161 

7263 

424 

7366 

7468 

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1038 

1139 

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1961 

2062 

2153 

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429 

2457 

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2963 

3064 

3165 

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430 

3468 

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4074 

4175 

4276 

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431 

4477 

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4880 

4981 

5081 

5182 

5283 

5383 

432 

5484 

5584 

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6785 

5886 

5986 

6087 

6187 

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433 

6488 

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6789 

6889 

6989 

7089 

7189 

7290 

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434 

7490 

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7790 

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8090 

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435 

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1177 

1276 

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438 

1474 

1673 

1672 

1771 

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1970 

2069 

2168 

2267 

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439 

2465 

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2761 

2860 

2959 

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3156 

3256 

3364 

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7187 

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7969 

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8165 

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8360 

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9140 

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9432 

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9627 

9724 

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650308 

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0890 

0987 

1084 

1181 

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1278 

1375 

1472 

1669 

1666 

1762 

1859 

1956 

2063 

2160 

449 

2246 

2343 

2440 

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2633 

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2826 

2923 

3019 

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LOGARITHMS 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

450 

653213 

3309 

3405 

3502 

3598 

3695 

3791 

3888 

3984 

4080 

451 

4177 

4273 

4369 

4465 

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4850 

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452 

5138 

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5427 

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5715 

5810 

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6002 

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6098 

6194 

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6386 

6482 

6577 

6673 

6769 

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454 

7056 

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7247 

7343 

7438 

7534 

7629 

7725 

7820 

7916 

455 

8011 

8107 

8202 

8298 

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8488 

8584 

8679 

8774 

8870 

456 

8965 

9060 

9155 

9250 

9346 

9441 

9536 

9631 

9726 

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9916 

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.581 

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458 

660865 

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1150 

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1339 

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1529 

1623 

1718 

459 

1813 

1907 

2002 

2096 

2191 

2286 

2380 

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2669 

2663 

460 

2758 

2852 

2947 

3041 

3135 

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3701 

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3889 

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4172 

4266 

4360 

4464 

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4642 

4736 

4830 

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6487 

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5581 

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5966 

6050 

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6331 

6424 

464 

6518 

6612 

6705 

6799 

6892 

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7079 

7173 

7266 

7360 

465 

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7546 

7640 

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8013 

8106 

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466 

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1080 

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1173 

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1636 

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1913 

2005 

470 

2098 

2190 

2283 

2375 

2467 

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2652 

2744 

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1693 

1784 

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1964 

2055 

481 

2146 

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2326 

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2777 

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2957 

482 

3047 

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4037 

4127 

4217 

4307 

4396 

4486 

4576 

4666 

4756 

484 

4854 

4935 

5026 

6114 

5204 

5294 

6383 

6473 

5563 

5652 

485 

5742 

5831 

5921 

6010 

6100 

6189 

6279 

6368 

6468 

6547 

486 

6636 

6726 

6816 

6904 

6994 

7083 

7172 

7261 

7361 

7440 

487 

7529 

7618 

7707 

7796 

7886 

7975 

8064 

8153 

8242 

8331 

488 

8420 

8509 

8598 

8687 

8776 

8865 

8953 

9042 

9131 

9220 

489 

9309 

9398 

9486 

9576 

9664 

9753 

9841 

9930 

..19 

.107 

490 

690196 

0285 

0373 

0362 

0550 

0639 

0728 

0816 

0905 

0993 

491 

1081 

1170 

1268 

1347 

1435 

1624 

1612 

1700 

1789 

1877 

492 

1965 

2053 

2142 

2230 

2318 

2406 

2494 

2583 

2671 

2759 

493 

2847 

2935 

3023 

3111 

3199 

3287 

3375 

3463 

3551 

3639 

494 

3727 

3816 

3903 

3991 

4078 

4166 

4264 

4342 

4430 

4517 

495 

4605 

4693 

4781 

4868 

4956 

5044 

6131 

6210 

5307 

5394 

496 

5482 

5569 

5657 

5744 

5832 

6919 

6007 

6094 

6182 

6269 

497 

6356 

5444 

6631 

6618 

6706 

6793 

6880 

6968 

7055 

7142 

498 

7229 

7317 

7404 

7491 

7578 

7665 

7752 

7839 

7926 

8014 

499 

8101 

8188 

8276 

8362 

8449 

8536 

8622 

8709 

8796 

8883 

OF  NUMBERS.             11 

N. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

500 

698970 

9057 

9144 

9231 

9317 

9404 

9491 

9678 

9664 

9751 

501 

9838 

9924 

..11 

..98 

.184 

.271 

.358 

.444 

.531 

.617 

602 

700704 

0790 

0877 

0963 

1050 

1136 

1222 

1309 

1395 

1482 

503 

1668 

1654 

1741 

1827 

1913 

1999 

2086 

2172 

2258 

2344 

504 

2431 

2517 

2603 

2t89 

2775 

2861 

2947 

3033 

3119 

3205 

506 

3291 

3377 

3463 

3549 

3635 

3721 

3807 

3895 

3979 

4065 

508 

4161 

4236 

4322 

4408 

4494 

4579 

4665 

4751 

4837 

4922 

507 

6008 

5094 

5179 

5265 

5350 

5436 

5522 

6607 

5693 

5778 

508 

6864 

6949 

6035 

6120 

6206 

6291 

6376 

6462 

6547 

6632 

509 

6718 

6803 

6888 

6974 

7059 

7144 

7229 

7316 

7400 

7485 

510 

7570 

7655 

7740 

7826 

7910 

7996 

8081 

8166 

8251 

8336 

511 

8421 

8606 

8591 

8676 

8761 

8846 

8931 

9015 

9100 

9185 

512 

9270 

9365 

9440 

9624 

9609 

9694 

9779 

9863 

9948 

..33 

513 

710117 

0202 

0287 

0371 

0466 

0540 

0625 

0710 

0794 

0879 

514 

0963 

1048 

1132 

1217 

1301 

1386 

1470 

1654 

1639 

1723 

515 

1807 

1892 

1976 

2060 

2144 

2229 

2313 

2397 

2481 

2666 

516 

2660 

2734 

2818 

2902 

2986 

3070 

3154 

3238 

3326 

3407 

517 

3491 

3675 

3659 

3742 

3826 

3910 

3994 

4078 

4162 

4246 

518 

4330 

4414 

4497 

4681 

4665 

4749 

4833 

4916 

5000 

5084 

519 

5167 

6261 

5335 

5418 

5502 

5586 

6669 

5753 

5836 

6920 

520 

6003 

6087 

6170 

6254 

6337 

6421 

6504 

6588 

6671 

6754 

521 

6838 

6921 

7004 

7088 

7171 

7254 

7338 

7421 

7504 

7587 

522 

7671 

7754 

7837 

7920 

8003 

8086 

8169 

8253 

8336 

8419 

523 

8602 

8585 

8668 

8751 

8834 

8917 

9000 

9083 

9166 

9248 

524 

9331 

9414 

9497 

9580 

9663 

9745 

9828 

9911 

9994 

..77 

525 

720169 

0242 

0325 

0407 

0490 

0573 

0656 

0738 

0821 

0903 

526 

0986 

1068 

1151 

1233 

1316 

1398 

1481 

1563 

1646 

1728 

527 

1811 

1893 

.976 

2058 

2140 

2222 

2305 

2387 

2469 

2652 

528 

2634 

2716 

2798 

2881 

2963 

3045 

3127 

3209 

3291 

3374 

529 

3466 

3638 

3620 

3702 

3784 

3866 

3948 

4030 

4112 

4194 

530 

4276 

4358 

4440 

4622 

4604 

4685 

4767 

4849 

4931 

5013 

631 

5095 

5176 

6258 

5340 

5422 

5503 

5585 

5667 

5748 

5830 

532 

5912 

5993 

6075 

6156 

6238 

6320 

6401 

6483 

6564 

6646 

533 

6727 

6809 

6890 

6972 

7053 

7134 

7216 

7297 

7379 

7460 

634 

7641 

7623 

7704 

7785 

7866 

7948 

8029 

8110 

8191 

8273 

535 

8354 

8436 

8516 

8597 

8678 

8759 

8841 

8922 

9003 

9084 

636 

9165 

9246 

9327 

9403 

9489 

9570 

9651 

9732 

9813 

9893 

637 

9974 

..66 

.136 

.217 

.298 

.378 

.459 

.440 

.621 

.702 

638 

730782 

0863 

0944 

1024 

1105 

1186 

1266 

1347 

1428 

1608 

539 

1689 

1669 

1760 

1830 

1911 

1991 

2072 

2162 

2233 

2313 

540 

2394 

2474 

2655 

2635 

2715 

2796 

2876 

2956 

3037 

3117 

541 

3197 

3278 

3358 

3438 

3518 

3598 

3679 

3759 

3839 

3919 

642 

3999 

4079 

4160 

4240 

4320 

4400 

4480 

4560 

4640 

4720 

643 

4800 

4^80 

4960 

5040 

6120 

5200 

5279 

5359 

5439 

5519 

644 

5599 

5679 

5759 

5838 

5918 

5998 

6078 

6167 

6237 

6317 

545 

6397 

6476 

6566 

6636 

6715 

6795 

6874 

6954 

7034 

7113 

646 

7193 

7272 

7352 

7431 

7511 

7590 

7670 

7749 

7829 

7908 

647 

7987 

8067 

8146 

8225 

8305 

8384 

8463 

8543 

8622 

8701 

548 

8781 

8860 

8939 

9018 

9097 

9177 

9256 

9335 

9414 

9493 

549 

9672 

9661 

9731 

9810 

9889 

9968 

..47 

.126 

.205 

.284 

12 

LOGARITHMS 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

550 

740363 

0442 

0521 

0560 

0678 

0757 

0836 

0915 

0994 

1073 

561 

1152 

1230 

1309 

1388 

1467 

1546 

1624 

1703 

1782 

1860 

552 

1939 

2018 

2096 

2175 

2254 

2332 

2411 

2489 

2568 

2646 

553 

2725 

2804 

2882 

2961 

3039 

3118 

3196 

3275 

3353 

3431 

554 

3510 

3558 

3667 

3745 

3823 

3902 

3980 

4058 

4136 

4215 

555 

4293 

4371 

4449 

4528 

4606 

4684 

4762 

4840 

4919 

4997 

556 

5075 

5153 

5231 

5309 

5387 

5465 

5543 

5621 

5699 

5777 

557 

5855 

5933 

6011 

6089 

6167 

6245 

6323 

6401 

6479 

6556 

558 

6634 

6712 

6790 

6868 

6945 

7023 

7101 

7179 

7256 

7334 

559 

7412 

7489 

7567 

7645 

7722 

7800 

7878 

7955 

8033 

8110 

560 

8188 

8266 

8343 

8421 

8498 

8576 

8653 

8731 

8808 

8885 

561 

8963 

9040 

9118 

9195 

9272 

9350 

9427 

9504 

9582 

9659 

562 

9736 

9814 

9891 

9968 

..45 

.123 

.200 

.277 

.354 

.431 

563 

750508 

0586 

0663 

0740 

0817 

0894 

0971 

1048 

1125 

1202 

564 

1279 

1356 

1433 

1510 

1587 

1664 

1741 

1818 

1895 

1972 

565 

2048 

2125 

2202 

2279 

2356 

2433 

2509 

2586 

2663 

2740 

566 

2816 

2893 

2970 

3047 

3123 

3200 

3277 

3353 

3430 

3506 

567 

3582 

3660 

3736 

3813 

3889 

3966 

4042 

4119 

4195 

4272 

568 

4348 

4425 

4501 

4578 

4654 

4730 

4807 

4883 

4960 

5036 

569 

5112 

5189 

5265 

5341 

5417 

5494 

5570 

5646 

5722 

5799 

570 

5875 

5951 

6027 

6103 

61F0 

6256 

6332 

6408 

6484 

6560 

571 

6636 

6712 

6788 

6864 

6940 

7016 

7092 

7168 

7244 

7320 

572 

7396 

7472 

7548 

7624 

7700 

7775 

7851 

7927 

8003 

8079 

573 

8155 

8230 

8306 

8382 

8458 

8533 

8609 

8685 

8761 

8836 

574 

8912 

8988 

9068 

9139 

9214 

9290 

9366 

9441 

9517 

9592 

575 

9668 

9743 

9819 

9894 

9970 

..45 

.121 

.196 

.272 

.347 

576 

760422 

0498 

0573 

0649 

0724 

0799 

0875 

0950 

1025 

1101 

577 

1176 

1251 

1326 

1402 

1477 

1552 

1627 

1702 

1778 

1853 

578 

1928 

2003 

2078 

2153 

2228 

2303 

2378 

2453 

2529 

2604 

579 

2679 

2754 

2829 

2904 

2978 

3053 

3128 

2203 

3278 

3353 

580 

3428 

3503 

3578 

3653 

3727 

3802 

3877 

3952 

4027 

4101 

581 

4176 

4251 

4326 

4400 

4475 

4550 

4624 

4699 

4774 

4848 

582 

4923 

4998 

5072 

5147 

5221 

5296 

5370 

5446 

5520 

6594 

583 

5669 

5743 

5818 

5892 

5966 

6041 

6115 

6190 

0264 

6338 

584 

6413 

6487 

6562 

6636 

6710 

6785 

6859 

6933 

7007 

7082 

585 

7156 

7230 

7304 

7379 

7453 

7527 

7601 

7675 

7749 

7823 

586 

7898 

7972 

8046 

8120 

8194 

8268 

8342 

8416 

8490 

8564 

587 

8638 

8712 

8786 

8860 

8934 

9008 

9082 

9156 

9230 

9303 

588 

9377 

9451 

9525 

9599 

9673 

9746 

9820 

9894 

9968 

..42 

589 

770115 

0189 

0263 

0336 

0410 

0484 

0557 

0631 

0705 

0778 

590 

0852 

0926 

0999 

1073 

1146 

1220 

1293 

1367 

1440 

1514 

591 

1587 

1661 

1734 

1808 

1881 

1955 

2028 

2102 

2175 

2248 

592 

2322 

2395 

2468 

3542 

2615 

2688 

2762 

2835 

2908 

2981 

593 

3055 

3128 

3201 

3274 

3348 

3421 

3494 

3567 

3640 

3713 

594 

3786 

3860 

3933 

4006 

4079 

4152 

4225 

4298 

4371 

4444 

595 

4517 

4590 

4663 

4736 

4809 

4882 

4955 

5028 

5100 

6173 

596 

5246 

5319 

5392 

5465 

5538 

5610 

5683 

5756 

5829 

5902 

597 

5974 

6047 

6120 

6193 

6265 

6338 

6411 

6483 

6566 

6629 

598 

6701 

6774 

6846 

6919 

6992 

7064 

7137 

7209 

7282 

7354 

599 

7427 

7499 

7572 

7644 

7717 

7789 

7862 

7934 

8006 

8079 

OF  NUMBERS.              13 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

600 

778151 

8224 

8296 

8368 

8441 

8513 

8585 

8658 

8730 

8802 

601 

8874 

8947 

9019 

9091 

9163, 

9236 

9308 

9380 

9452 

9624 

602 

9596 

6669 

9741 

9813 

9885 

9967 

..29 

.101 

.173 

.245 

603 

780317 

0389 

0461 

0533 

0605 

0677 

0749 

0821 

0893 

0965 

604 

1037 

1109 

1181 

1253 

1324 

1396 

1468 

1540 

1612 

1684 

605 

1755 

1827 

1899 

1971 

2042 

2114 

2186 

2258 

2329 

2401 

606 

2473 

2644 

2616 

2688 

2769 

2831 

2902 

2974 

3046 

3117 

607 

3189 

3260 

3332 

3403 

3475 

3646 

3618 

3689 

3761 

3832 

608 

3904 

3975 

4046 

4118 

4189 

4261 

4332 

4403 

4475 

4646 

609 

4617 

4689 

4760 

4831 

4902 

4974 

5045 

5116 

5187 

5259 

610 

5330 

5401 

5472 

5543 

5615 

5686 

5757 

5828 

5899 

5970 

611 

6041 

6112 

6183 

6254 

6325 

6396 

6467 

6638 

6609 

6680 

612 

6751 

6822 

6893 

6964 

7035 

7106 

7177 

7248 

7319 

7390 

613 

7460 

7631 

7602 

7673 

7744 

7815 

7885 

7956 

8027 

8098 

614 

8168 

8239 

8310 

8381 

8451 

8522 

8593 

8663 

8734 

8804 

615 

8875 

8946 

9016 

9087 

9157 

9228 

9299 

9369 

9440 

9510 

616 

9581 

9651 

9722 

9792 

9863 

9933 

...4 

..74 

.144 

.216 

617 

790285 

0356 

0426 

0496 

0667 

0637 

0707 

0778 

0848 

0918 

618 

0988 

1069 

1129 

1199 

1269 

1340 

1410 

1480 

1560 

1620 

619 

1691 

1761 

1831 

1901 

1971 

2041 

2111 

2181 

2252 

2322 

620 

2392 

2462 

2532 

2602 

2672 

2742 

2812 

2882 

2952 

3022 

621 

3092 

3162 

3231 

3301 

3371 

3441 

3511 

3581 

3651 

3721 

622 

3790 

3860 

3930 

4000 

4070 

4139 

4209 

4279 

4349 

4418 

623 

4488 

4558 

4627 

4697 

4767 

4836 

4906 

4976 

5045 

6115 

624 

5185 

5254 

5324 

5393 

5463 

5532 

5602 

5672 

5741 

5811 

625 

5880 

5949 

6019 

6088 

6158 

6227 

6297 

6366 

6436 

6505 

626 

6574 

6644 

6713 

6782 

6862 

6921 

6990 

7060 

7129 

7198 

627 

7268 

7337 

7406 

7475 

7646 

7614 

7683 

7752 

7821 

7890 

628 

7960 

8029 

8098 

8167 

8236 

8305 

8374 

8443 

8613 

8582 

629 

8651 

8720 

8789 

8858 

8927 

8996 

9065 

6134 

9203 

9272 

630 

9341 

9409 

9478 

9547 

9610 

9685 

9754 

9823 

9892 

9961 

631 

800026 

0098 

0167 

0236 

0305 

0373 

0442 

0511 

0580 

0648 

632 

0717 

0786 

0854 

0923 

0992 

1061 

1129 

1198 

1266 

1335 

633 

1404 

1472 

1641 

1609 

1678 

1747 

1816 

1884 

1952 

2021 

634 

2089 

2158 

2226 

2295 

2363 

2432 

2500 

2568 

2637 

2705 

635 

2774 

2842 

2910 

2979 

3047 

3116 

3184 

3252 

3321 

3389 

636 

3467 

3626 

3594 

3662 

3730 

3798 

3867 

3935 

4003 

4071 

637 

4139 

4208 

4276 

4354 

4412 

4480 

4548 

4616 

4685 

4753 

638 

4821 

4889 

4967 

5026 

6093 

5161 

6229 

5297 

5366 

5433 

639 

5601 

5669 

5637 

5706 

5773 

5841 

6908 

5976 

6044 

6112 

640 

6180 

6248 

6316 

6384 

6451 

6519 

6587 

6655 

6723 

6790 

641 

6858 

6926 

6994 

7061 

7129 

7167 

7264 

7332 

7400 

7467 

642 

7535 

7603 

7670 

7738 

7806 

7873 

7941 

8008 

8076 

8143 

643 

8211 

8279 

8346 

8414 

8481 

8649 

8616 

8684 

8751 

8818 

644 

8886 

8953 

9021 

9088 

9166 

9223 

9290 

9358 

9425 

9492 

645 

9560 

9627 

9694 

9762 

9829 

9896 

9964 

..31 

..98 

.165 

646 

810233 

0300 

0367 

0434 

0501 

0596 

0636 

0703 

0770 

0837 

647 

0904 

0971 

1039 

1106 

1173 

1240 

1307 

1374 

1441 

1508 

648 

1576 

1642 

1709 

1776 

1843 

1910 

1977 

2044 

2111 

2178 

649 

2246 

2312 

2379 

2445 

2512 

2679 

2646 

2713 

2780 

2847 

14 

LOGARITHMS 

N. 

0 

I 
2980 

2 

3 

4 

5 

6 

7 

8 

9 

650 

812913 

3047 

3114 

3181 

3247 

3314 

3381 

3448 

3514 

651 

3581 

3648 

3714 

3781 

3848 

3914 

3981 

4048 

4114 

4181 

652 

4248 

4314 

4381 

4447 

4514 

4581 

4647 

4714 

4780 

4847 

653 

4913 

4980 

5046 

6113 

5179 

5246 

5312 

5378 

5446 

5611 

654 

5578 

5644 

5711 

5777 

6843 

5910 

5976 

6042 

6109 

6175 

655 

6241 

6308 

6374 

6440 

6506 

6573 

6639 

6705 

6771 

6838 

666 

6904 

6970 

7036 

7102 

7169 

7233 

7301 

7367 

7433 

7499 

657 

7565 

7631 

7698 

7764 

7830 

7896 

7962 

8028 

8094 

8160 

658 

8226 

8292 

8358 

8424 

8490 

8666 

8622 

8688 

8754 

8820 

659 

8885 

8951 

9017 

9083 

9149 

9215 

9281 

9346 

9412 

9478 

660 

9544 

9610 

9676 

9741 

9807 

9873 

9939 

...4 

..70 

.136 

661 

820201 

0267 

0333 

0399 

0464 

0630 

0595 

0661 

0727 

0792 

662 

0858 

0924 

0989 

1065 

1120 

1186 

1261 

1317 

1382 

1448 

663 

1514 

1579 

1645 

1710 

1775 

1841 

1906 

1972 

2037 

2103 

664 

2168 

2233 

2299 

2364 

2430 

2496 

2660 

2626 

2691 

2756 

665 

2822 

2887 

2952 

3018 

3083 

3148 

3213 

3279 

3344 

3409 

666 

3474 

3539 

3606 

3670 

3735 

3800 

3866 

3930 

3996 

4061 

667 

4126 

4191 

4256 

4321 

4386 

4461 

4516 

4681 

4646 

4711 

668 

4776 

4841 

4906 

4971 

5036 

5101 

5166 

6231 

5296 

5361 

669 

5426 

5491 

5566 

5621 

6686 

5761 

5815 

5880 

5945 

6010 

670 

6075 

6140 

6204 

6269 

6334 

6399 

6464 

6528 

6593 

6658 

671 

6723 

6787 

6862 

6917 

6981 

7046 

7111 

7175 

7240 

7305 

672 

7369 

7434 

7499 

7563 

7628 

7692 

7767 

7821 

7886 

7951 

673 

8015 

8080 

8144 

8209 

8273 

8338 

8402 

8467 

8631 

8595 

674 

8660 

8724 

8V89 

8853 

8918 

8982 

9046 

9111 

9176 

9239 

675 

9304 

9368 

9432 

9497 

9561 

9625 

9690 

9764 

9818 

9882 

676 

9947 

..11 

..75 

.139 

.204 

.268 

.332 

.396 

.460 

.625 

677 

830589 

0663 

0717 

0781 

0845 

0909 

0973 

1037 

1102 

1166 

678 

1230 

1294 

1358 

1422 

1486 

1660 

1614 

1678 

1742 

1806 

679 

1870 

1934 

1998 

2062 

2126 

2189 

2263 

2317 

2381 

2445 

680 

2509 

2573 

2637 

2700 

2764 

2828 

2892 

2956 

3020 

3083 

681 

3147 

3211 

3275 

3338 

3402 

3466 

3530 

3593 

3667 

3721 

682 

3784 

3848 

3912 

3976 

4039 

4103 

4166 

4230 

4294 

4357 

683 

4421 

4484 

1  4648 

4611 

4676 

4739 

4802 

4866 

4929 

4993 

684 

5056 

5120 

5183 

5247 

5310 

6373 

5437 

5500 

5664 

5627 

685 

5691 

6754 

!  5817 

5881 

5944 

6007 

6071 

6134 

6197 

6261 

686 

6324 

6387 

6461 

6514 

6677 

6641 

6704 

6767 

6830 

6894 

687 

6957 

7020 

7083 

7146 

7210 

7273 

7336 

7399 

7462 

7525 

688 

7688 

7652 

7716 

7778 

7841 

7904 

7967 

8030 

8093 

8166 

689 

8219 

8282 

8345 

8408 

8471 

8534 

8597 

8660 

8723 

8786 

690 

8849 

8912 

8975 

9038 

9109 

9164 

9227 

9289 

9352 

9415 

691 

9478 

9541 

9604 

9667 

9729 

9792 

9865 

9918 

9981 

..43 

692 

840106 

0169 

0232 

0294 

0367 

0420 

0482 

0546 

0608 

0671 

693 

0733 

0796  0859 

0921 

0984 

1046 

1109 

1172 

1234 

1297 

694 

1359 

1422  1485 

1647 

1610 

1672 

1735 

1797 

1860 

1922 

695 

1985 

2047  2110 

2172 

2235 

2297 

2360 

2422 

2484 

2547 

696 

2609 

2672  2734 

2796 

2869 

2921 

2983 

3046 

3108 

3170 

697 

3233 

3295  3357 

3420 

3482 

3644 

3606 

3669 

3731 

3793 

698 

3865 

3918  3980 

4042 

4104 

4166 

4229 

4291 

4353 

4415 

e. 

4477 

4539  4601 

4664 

1 

4726 

4788 

4860 

4912 

4974 

5036 

OF  NUMBERS.             15 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

700 

846098 

6160 

6222 

5284 

5346 

5408 

6470 

5532 

5694 

5656 

701 

5718 

5780 

6842 

5904 

5966 

6028 

6090 

6161 

6213 

6275 

702 

6337 

6399 

6461 

6523 

6586 

6646 

6708 

6770 

6832 

6894 

703 

6955 

7017 

7079 

7141 

7202 

7264 

7326 

7388 

7449 

7511 

704 

7573 

7634 

7676 

7768 

7819 

7831 

7943 

8004 

8066 

8128 

705 

8189 

8251 

8312 

8374 

8435 

8497 

8569 

8620 

8682 

8743 

706 

8805 

8866 

8928 

8989 

9051 

9112 

9174 

9235 

9297 

9358 

707 

9419 

9481 

9542 

9604 

9665 

9726 

9788 

9849 

9911 

9972 

708 

850033 

0095 

0156 

0217 

0279 

0340 

0401 

0462 

0524 

0586 

709 

0646 

0707 

0769 

0830 

0891 

0952 

1014 

1075 

1136 

1197 

710 

1258 

1320 

1381 

1442 

1603 

1564 

1625 

1686 

1747 

1809 

711 

1870 

1931 

1992 

2053 

2114 

2175 

2236 

2297 

2368 

2419 

712 

2480 

2541 

2602 

2663 

2724 

2785 

2846 

2907 

2968 

3029 

713 

3090 

3150 

3211 

3272 

3333 

3394 

3455 

3616 

3577 

3637 

714 

3698 

3759 

3820 

3881 

3941 

4002 

4063 

4124 

4185 

4245 

715 

4306 

4367 

4428 

4488 

4649 

4610 

4670 

4731 

4792 

4852 

716 

4913 

4974 

6034 

5095 

5156 

6216 

6277 

6337 

5398 

5459 

717 

5519 

5580 

5640 

5701 

5761 

5822 

6882 

5943 

6003 

6064 

718 

6124 

6185 

6246 

6306 

6366 

6427 

6487 

6548 

6608 

6668 

719 

6729 

6789 

6850 

6910 

6970 

7031 

7091 

7152 

7212 

7272 

720 

7332 

7393 

7453 

7513 

7574 

7634 

7694 

7755 

7815 

7875 

721 

7935 

7995 

8056 

8116 

8176 

8236 

8297 

8357 

8417 

8477 

722 

8537 

8597 

8657 

8718 

8778 

8838 

8898 

8958 

9018 

9078 

723 

9138 

9198 

9258 

9318 

9379 

9439 

9499 

9559 

9619 

9679 

724 

9739 

9799 

9859 

9918 

9978 

..38 

..98 

.168 

.218 

.278 

'725 

860338 

0398 

0458 

0618 

0578 

0637 

0697 

0757 

0817 

0877 

726 

0937 

0996 

1066 

1116 

1176 

1236 

1295 

1355 

1416 

1475 

727 

1634 

1594 

1654 

1714 

1773 

1833 

1893 

1952 

2012 

2072 

728 

2131 

2191 

2251 

2310 

2370 

2430 

2489 

2549 

2608 

2668 

729 

2728 

2787 

2847 

2906 

2966 

3025 

3086 

3144 

3204 

3263 

730 

3323 

3382 

3442 

3501 

3561 

3620 

3680 

3739 

3799 

3858 

731 

3917 

3977 

4036 

4096 

4166 

4214 

4274 

4333 

4392 

4452 

732 

4611 

4570 

4630 

4689 

4148 

4808 

4867 

4926 

4986 

5045 

733 

6104 

5163 

5222 

6282 

6341 

5400 

5459 

6619 

5678 

5637 

734 

6696 

5755 

6814 

6874 

5933 

5992 

6051 

6110 

6169 

6228 

735 

6287 

6346 

6405 

6466 

6524 

6583 

6642 

6701 

6760 

6819 

736 

6878 

6937 

6996 

7055 

7114 

7173 

7232 

7291 

7350 

7409 

737 

7467 

7526 

7685 

7644 

7703 

7762 

7821 

7880 

7939 

7998 

738 

8056 

8115 

8174 

8233 

8292 

8360 

8409 

8468 

8527 

8586 

739 

8644 

8703 

8762 

8821 

8879 

8938 

8997 

9056 

9114 

9173 

740 

9232 

9290 

9349 

9408 

9466 

9525 

9584 

9642 

9701 

9760 

741 

9818 

9877 

9936 

9994 

..53 

.111 

.170 

.228 

.287 

.345 

742 

870404 

0462 

0521 

0679 

0638 

0696 

0756 

0813 

0872 

0930 

743 

0989 

1047 

1106 

1164 

1223 

1281 

1339 

1398 

1456 

15i5 

744 

1573 

1631 

1690 

1748 

1806 

1866 

1923 

1981 

2040 

2098 

745 

2156 

2215 

2273 

2331 

2389 

2448 

2506 

2564 

2622 

2681 

746 

2739 

2797 

2855 

2913 

2972 

3030 

3088 

3146 

3204 

3262 

747 

3321 

3379 

3437 

3495 

3653 

3611 

3669 

3727 

3786 

3844 

748 

3902 

3960 

4018 

4076 

4134 

4192 

4260 

4308 

4360 

4424 

749 

4482 

4640 

4698 

4666 

4714 

4772 

4830 

4888 

4945 

5003 

16 

LOGARITHMS 

N. 

0 

1 

2 

3     4 

6 

6 

7 

8 

9 

750 

8750G1 

5119 

5177 

5235 

5293 

5351 

5409 

5466 

5524 

5582 

761 

5640 

5698 

5756 

6813 

5871 

5929 

6987 

6045 

6102 

6160 

752 

6218 

6276 

6333 

6391 

6449 

6507 

6564 

6622 

6680 

6737 

753 

6795 

6853 

6910 

6968 

7026 

7083 

7141 

7199 

7256 

7314 

754 

7371 

7429 

7487 

7544 

7602 

7659 

7717 

7774 

7832 

7889 

755 

7947 

8004 

8062 

8119 

8177 

8234 

8292 

8349 

8407 

8464 

756 

8522 

8579 

8637 

8694 

8762 

8809 

8886 

8924 

8931 

9039 

757 

9096 

9153 

9211 

9268 

9325 

9383 

9440 

9497 

9555 

9612 

758 

9669 

9726 

9784 

9841 

9898 

9956 

..13 

..70 

.127 

.185 

759 

880242 

0299 

0356 

0413 

0471 

0528 

0580 

0642 

0699 

0756 

760 

0814 

0871 

0928 

0985 

1042 

1099 

1156 

1213 

1271 

1328 

761 

1385 

1442 

1499 

1656 

1613 

1670 

1727 

1784 

1841 

1898 

762 

1955 

2012 

2069 

2126 

2183 

2240 

2297 

2354 

2411 

2468 

763 

2525 

2581 

2638 

2696 

2762 

2809 

2866 

2923 

2980 

3037 

764 

3093 

3150 

3207 

3264 

3321 

3377 

3434 

3491 

3548 

3606 

765 

3661 

3718 

3775 

3832 

3888 

3945 

4002 

4059 

4115 

4172 

766 

4229 

4285 

4342 

4399 

4455 

4512 

4569 

4626 

4682 

4739 

767 

4795 

4852 

4909 

4965 

6022 

5078 

5136 

6192 

5248 

5305 

768 

6361 

5418 

5474 

5631 

6587 

6644 

6700 

5757 

5813 

5870 

769 

5926 

5983 

6039 

6096 

6152 

6209 

6265 

6321 

6378 

6434 

770 

6491 

6547 

6604 

6660 

6716 

6773 

6829 

6885 

6942 

6998 

771 

7054 

7111 

7167 

7233 

7280 

7336 

7392 

7449 

7606 

7561 

772 

7617 

7674 

7730 

7786 

7842 

7898 

7956 

8011 

8067 

8123 

773 

8179 

8236 

8292 

8348 

8404 

8460 

8616 

8573 

8629 

8666 

774 

8741 

8797 

8853 

8909 

8965 

9021 

9077 

9134 

9190 

9246 

775 

9302 

9358 

9414 

9470 

9626 

9582 

9638 

9694 

9760 

9806 

776 

9862 

9918 

0974 

..30 

..86 

.141 

.197 

.253 

.309 

.365 

777 

890421 

0477 

0533 

0589 

0646 

0700 

0756 

0812 

0868 

0924 

778 

0980 

1035 

1091 

1147 

1203 

1259 

1314 

1370 

1426 

1482 

779 

1537 

1693 

1649 

1705 

1760 

1816 

1872 

1928 

1983 

2039 

780 

2095 

2150 

2206 

2262 

2317 

2373 

2429 

2484 

2540 

2595 

781 

2651 

2707 

2762 

3818 

2873 

2929 

2985 

3040 

3096 

3161 

782 

3207 

3262 

3318 

3373 

3429 

3484 

3540 

3595 

3651 

3706 

783 

3762 

3817 

3873 

3928 

3984 

4039 

4094 

4150 

4205 

4261 

784 

4316 

4371 

4427 

4482 

4638 

4593 

4648 

4704 

4759 

4814 

785 

4870 

4926 

4980 

5036 

5091 

5146 

6201 

5267 

5312 

6367 

786 

5423 

5478 

5533 

5688 

5644 

5699 

5754 

6809 

5864 

6920 

787 

5976 

6030 

6085 

6140 

6196 

6261 

6306 

6361 

6416 

6471 

788 

6526 

6581 

6636 

6692 

6747 

6802 

6867 

6912 

6967 

7022 

789 

7077 

7132 

7187 

7242 

7297 

7362 

7407 

7462 

7517 

7672 

790 

7627 

7683 

7737 

7792 

7847 

7902 

7967 

8012 

8067 

8122 

791 

8176 

8231 

8286 

8341 

8396 

8451 

8506 

8661 

8615 

8670 

792 

8725 

8780 

8836 

8890 

8944 

8999 

9054 

9109 

9164 

9218 

793 

9273 

9328 

9383 

9437 

9492 

9647 

9602 

9656 

9711 

9766 

794 

9821 

9875 

9930 

9986 

..39 

..94 

.149 

.203 

.258 

.312 

795 

900367 

0422 

0476 

0631 

0586 

0640 

0695 

0749 

0804 

0869 

796 

0913 

0968 

1022 

1077 

1131 

1186 

1240 

1295 

1349 

1404 

797 

1458 

1513 

1567 

1622 

1676 

1736 

1785 

1840 

lfc94 

1948 

798 

2003 

2057 

2112 

2166 

2221 

2275 

2329 

2384 

2438 

2492 

799 

2547 

2601 

2656 

2710  2764 

2818 

2873 

2927 

2981 

3036 

OF  NUMBERS.              17 

N. 

0 

1 

2     3 

4 

5 

6 

7 

8 

9 

800 

903090 

3144 

3199 

3253 

3307 

3361 

3416 

3470 

3524 

3578 

801 

3633 

3687 

3741 

3795 

3849 

3904 

3968 

4012 

4066 

4120 

802 

4174 

4229 

4283 

4337 

4391 

4445 

4499 

4553 

4607 

4661 

803 

4716 

4770 

4824 

4878 

4932 

4986 

5040 

5094 

5148 

6202 

804 

5256 

5310 

5364 

5418 

5472 

5526 

5580 

5634 

5688 

5742 

805 

5796 

5850 

5904 

5958 

6012 

6066 

6119 

6173 

6227 

6281 

806 

6335 

6389 

6443 

6497 

6551 

6604 

6658 

6712 

6766 

6820 

807 

6874 

6927 

6981 

7035 

7089 

7143 

7196 

7250 

7304 

7358 

808 

7411 

7465 

7619 

7573 

7626 

7680 

7734 

7787 

7841 

7896 

809 

7949 

8002 

8056 

8110 

8163 

8217 

8270 

8324 

8378 

8431 

810 

8485 

8539 

8592 

8646 

8699 

8763 

8807 

8860 

8914 

8967 

811 

9021 

9074 

9128 

9181 

9236 

9289 

9342 

9396 

9449 

9603 

812 

9556 

9610 

9663 

9716 

9770 

9823 

9877 

9930 

9984 

..37 

813 

910091 

0144 

^197 

0251 

0304 

0358 

0411 

0464 

0518 

0571 

814 

0624 

0878 

0731 

0784 

0838 

0891 

0944 

0998 

1051 

1104 

815 

1158 

1211 

1264 

1317 

1371 

1424 

1477 

1530 

1584 

1637 

816 

1690 

1743 

1797 

1850 

1903 

1956 

2009 

2063 

2115 

2169 

817 

2222 

2275 

2323 

2381 

2435 

2488 

2541 

2594 

2645 

2700 

818 

2753 

2806 

2859 

2913 

2966 

3019 

3072 

3125 

3178 

3231 

819 

3284 

3337 

3390 

3443 

3496 

3649 

3602 

3656 

3708 

3761 

820 

3814 

3867 

3920 

3973 

4026 

4079 

4132 

4184 

4237 

4290 

821 

4343 

4396 

4449 

4502 

4555 

4608 

4660 

4713 

4766 

4819 

822 

4872 

4925 

4977 

5030 

5083 

5136 

5189 

5241 

5594 

6347 

823 

5400 

5453 

5505 

5558 

5611 

5664 

5716 

6769 

5822 

5875 

824 

5927 

6980 

6033 

6085 

6138 

6191 

6243 

6296 

6349 

6401 

825 

6454 

6507 

6559 

6612 

6664 

6717 

6770 

6822 

6875 

6927 

826 

6980 

7033 

7085 

7138 

7190 

7243 

7295 

7348 

7400 

7453 

827 

7606 

7558 

7611 

7663 

7716 

7768 

7820 

7873 

7925 

7978 

828 

8030 

8083 

8185 

8188 

8240 

8293 

8345 

8397 

8450 

8602 

829 

8555 

8607 

8659 

8712 

8764 

8816 

8869 

8921 

8973 

9026 

830 

9078 

9130 

9183 

9235 

9287 

9340 

9392 

9444 

9496 

9549 

831 

9601 

9653 

9706 

9758 

9810 

9862 

9914 

9967 

..19 

..71    ! 

832 

920123 

0176 

0228 

0280 

0332 

0384 

0436 

0489 

0641 

0693 
1114 

833 

0645 

0897 

0749 

0801 

0863 

0906 

0958 

1010 

1062 

834 

1166 

1218 

1270 

1322 

1374 

1426 

1478 

1530 

1682 

1634 

835 

1686 

1738 

1790 

1842 

1894 

1946 

1998 

2060 

2102 

2154 

836 

2206 

2258 

2310 

2362 

2414 

2466 

2618 

2570 

2622 

2674 

837 

2725 

2777 

2829 

2881 

2933 

2985 

3037 

3089 

3140 

3192 

838 

3244 

3296 

3348 

3399 

3451 

3603 

3556 

3607 

3668 

3710 

839 

3762 

3814 

3865 

3917 

3969 

4021 

4072 

4124 

4147 

4228 

840 

4279 

4331 

4383 

4434 

4486 

4538 

4589 

4641 

4693 

4744 

841 

4796 

4848 

4899 

4951 

5003 

6054 

6106 

5157 

6209 

5261 

842 

5312 

6364 

5415 

6467 

6518 

6570 

5621 

5673 

6725 

5776 

843 

5828 

5874 

6931 

5982 

6034 

6086 

6137 

6188 

6240 

6291 

844 

6342 

6394 

6445 

6497 

6548 

6600 

6651 

6702 

6754 

6805 

845 

6857 

6908 

6959 

7011 

7062 

7114 

7165 

7216 

7268 

7319 

846 

7370 

7422 

7473 

7524 

7576 

7627 

7678 

7730 

7783 

7832 

847 

7883 

7935 

7986 

8037 

8088 

8140 

8191 

8242 

8293 

8345 

848 

8396 

8447 

8498 

8549 

8601 

8652 

8703 

8754 

8805 

8857 

849 

8908 

8959 

9010 

9061 

9112 

9163 

9216 

9266 

9317 

9368 

18 

LOGARITHMS 

N. 

0 
929419 

1  1  3 

3 

4 

5 

6 

7 

8 

9 

850 

9473  9521 

9672 

9623 

9674 

9725 

9776 

9827 

9879 

851 

9930 

9981 

..32 

..83 

.134 

.185 

.236 

.287 

.338 

.389 

852 

930440 

0491 

0542 

0592 

0643 

0694 

0745 

0796 

0847 

0898 

853 

0949 

1000 

1051 

1102 

1163 

1204 

1254 

1305 

1356 

1407 

854 

1458 

1609 

1560 

1610 

1661 

1712 

1763 

1814 

1865 

1915 

855 

1966 

2017 

2068 

2118 

2169 

2220 

2271 

2322 

2372 

2423 

856 

2474 

2624 

2676 

2626 

2677 

2727 

2778 

2829 

2879 

2930 

857 

2981 

3031 

3082 

3133 

3183 

3234 

3285 

3335 

3386 

3437 

858 

3487 

3538 

3589 

3639 

3690 

3740 

3791 

3841 

3892 

3943 

859 

3993 

4044 

4094 

4145 

4196 

4246 

4269 

4347 

4397 

4448 

860 

4498 

4549 

4599 

4650 

4700 

4751 

4801 

4852 

4902 

4953 

861 

5003 

6054 

5104 

5154 

5205 

6255 

6306 

535t? 

6406 

5467 

862 

6507 

5558 

6608 

5658 

6709 

5769 

5809 

68G0 

5910 

5960 

863 

6011 

6061 

6111 

6162 

6212 

6262 

6313 

6363 

6413 

6463 

864 

6514 

6664 

6614 

6665 

6715 

6765 

6816 

6865 

6916 

6966 

865 

7016 

7066 

7117 

7167 

7217 

7267 

7317 

7367 

7418 

7468 

866 

7518 

7568 

7618 

7668 

7718 

7769 

7819 

7869 

7919 

7969 

867 

8019 

8069 

8119 

8169 

8219 

8269 

8320 

8370 

8420 

8470 

868 

8620 

8570 

8620 

8670 

8720 

8770 

8820 

8870 

8919 

8970 

869 

9020 

9070 

9120 

9170 

9220 

9270 

9320 

9369 

9419 

9469 

870 

9519 

9569 

9616 

9669 

9719 

9769 

9819 

9869 

9918 

9968 

871 

940018 

0068 

0118 

0168 

0218 

0267 

0317 

0367 

0417 

0467 

872 

0616 

0566 

0616 

0666 

0716 

0765 

0815 

0866 

0916 

0964 

873 

1014 

1064 

1114 

1163 

1213 

1263 

1313 

1362 

1412 

1462 

874 

1611 

1561 

1611 

1660 

1710 

1760 

1809 

1869 

1909 

1958 

875 

2008 

2058 

2107 

2157 

2207 

2256 

2306 

2355 

2405 

2455 

876 

2504 

2664 

2603 

2653 

2702 

2762 

2801 

2851 

2901 

2950 

877 

3000 

3049 

3099 

3148 

3198 

3247 

3297 

3346 

3396 

3445 

878 

3495 

3644 

3693 

3643 

3692 

3742 

3791 

3841 

3890 

3939 

879 

3989 

4038 

4088 

4137 

4186 

4236 

4285 

4336 

4384 

4433 

880 

4483 

4632 

4581 

4631 

4680 

4729 

4779 

4828 

4877 

4927 

881 

4976 

5025 

5074 

5124 

6173 

6222 

5272 

5321 

6370 

5419 

882 

6469 

5618 

5667 

6616 

5665 

6715 

5764 

6813 

6862 

5912 

883 

6961 

6010 

6059 

6108 

6157 

6207 

6256 

6305 

6354 

6403 

884 

6462 

6501 

6661 

6600 

6649 

6698 

6747 

6796 

6845 

6894 

885 

6943 

6992 

7041 

7090 

7140 

7189 

7238 

7287 

7336 

7385 

886 

7434 

7483 

7532 

7581 

7630 

7679 

7728 

7777 

7826 

7875 

887 

7924 

7973 

8022 

8070 

8119 

8168 

8217 

8266 

8315 

8365 

888 

8413 

8462 

8511 

8560 

8609 

8667 

8706 

8755 

8804 

8a53 

889 

8902 

8961 

8999 

9048 

9097 

9146 

9195 

9244 

9:^y2 

9341 

890 

9390 

9439 

9488 

9636 

9585 

9634 

9683 

9731 

9780 

9829 

891 

9878 

9926 

9976 

..24 

..73 

.121 

.170 

.219 

.267 

.316 

892 

950365 

0414 

0462 

0511 

0560 

0608 

0657 

0706 

0754 

0803 

893 

0851 

0900 

0949 

0997 

1046 

1096 

1143 

1192 

1240 

1289 

894 

1338 

1386 

1435 

1483 

1532 

1580 

1629 

1677 

1726 

17.5 

895 

1823 

1872 

1920 

1969 

2017 

2066 

2114 

2163 

2211 

2260 

896 

2308 

2356 

2405 

2453 

2602 

2550 

2599 

2647 

6696 

2744 

897 

2792 

2841 

2889 

2938 

2986 

3034 

3083 

3131 

3180 

3228 

898 

3276 

3325 

3373 

3421 

3470 

3518 

3566 

3615 

3663 

3711 

899 

3760 

3808 

3856 

3905 

3963 

4001 

4049 

4098 

4146 

4194 

OF  NUMBERS.             19 

N. 

0 

1 

2 

3 

4 

« 

6 

7 

8 

9 

900 

954243 

4291 

4339 

4387 

4435 

4484 

4532 

4580 

4628 

4677 

901 

4725 

4773 

4821 

4869 

4918 

4966 

5014 

5062 

5110 

5158 

902 

5207 

5255 

5303 

5351 

5399 

5447 

5495 

5543 

5592 

5640 

903 

5688 

5736 

5784 

5832 

5880 

5928 

5976 

6024 

6072 

6120 

904 

6168 

6216 

6265 

6313 

6361 

6409 

6457 

6505 

6553 

6601 

905 

6649 

6697 

6745 

6793 

6840 

6888 

6936 

6984 

7032 

7080 

906 

7128 

7176 

7224 

7272 

7320 

7368 

7416 

7464 

7612 

7659 

907 

7607 

7655 

7703 

7751 

7799 

7847 

7894 

7942 

7990 

8038 

908 

8086 

8134 

8181 

8229 

8277 

8325 

8373 

8421 

8468 

8516 

909 

8564 

8612 

8659 

8707 

8755 

8803 

8850 

8898 

8946 

8994 

910 

9041 

9089 

9137 

9185 

9232 

9280 

9328 

9375 

9423 

9471 

911 

9518 

9566 

9614 

9661 

9709 

9757 

9804 

9852 

9900 

9947 

912 

9995 

..42 

..90 

.138 

.185 

.233 

.280 

.328 

.376 

.423 

913 

960471 

0518 

0566 

0613 

0661 

0709 

0756 

0804 

0851 

0899 

914 

0946 

0994 

1041 

1089 

U36 

1184 

1231 

1279 

1326 

1374 

915 

1421 

1469 

1516 

1563 

1611 

1658 

1706 

1753 

1801 

1848 

916 

1895 

1943 

1990 

2038 

2085 

2132 

2180 

2227 

2275 

2322 

917 

2369 

2417 

2464 

2511 

2559 

2606 

2653 

2701 

2748 

2795 

918 

2843 

2890 

2937 

2985 

3032 

3079 

3126 

3174 

3221 

3268 

919 

3316 

3363 

3410 

3457 

3604 

3552 

3599 

3646 

3693 

3741 

920 

3788 

3835 

3882 

3929 

3977 

4024 

4071 

4118 

4165 

4212 

921 

4260 

4307 

4354 

4401 

4448 

4495 

4542 

4590 

4637 

4684 

922 

4731 

4778 

4825 

4872 

4919 

4966 

5013 

5061 

5108 

6155 

923 

5202 

5249 

5296 

5343 

5390 

5437 

5484 

5531 

5578 

5626 

924 

5672 

5719 

5766 

5813 

5860 

6907 

5964 

6001 

6048 

6095 

'925 

6142 

6189 

6236 

6283 

6329 

6376 

6423 

6470 

6517 

6564 

926 

6611 

6658 

6705 

6752 

6799 

6845 

6892 

6939 

6986 

7033 

927 

7080 

7127 

7173 

7220 

7267 

7314 

7361 

7408 

7464 

7501 

928 

7548 

7595 

7642 

7688 

7735 

7782 

7829 

7875 

7922 

7969 

929 

8016 

8062 

8109 

8156 

8203 

8249 

8296 

8343 

8390 

8436 

930 

8483 

8530 

8576 

8623 

8670 

8716 

8763 

8810 

8856 

8903 

931 

8950 

8996 

9043 

9090 

9136 

9183 

9229 

9276 

9323 

9369 

932 

9416 

9463 

9509 

9556 

9602 

9649 

9695 

9742 

9789 

9835 

933 

9882 

9928 

9975 

..21 

..68 

.114 

.161 

.207 

.254 

.300 

934 

970347 

0393 

0440 

0486 

0533 

0579 

0626 

0672 

0719 

0765 

935 

0812 

0858 

0904 

0951 

0997 

1044 

1090 

1137 

1183 

1229 

936 

1276 

1322 

1369 

1415 

1461 

1508 

1554 

1601 

1647 

1693 

937 

1740 

1786 

1832 

1879 

1925 

1971 

2018 

2064 

2110 

2157 

938 

2203 

2249 

2295 

2342 

2388 

2434 

2481 

2527 

2573 

2619 

939 

2666 

2712 

2758 

2804 

2851 

2897 

2943 

2989 

3035 

3082 

940 

3128 

3174 

3220 

3266 

3313 

3359 

3405 

3451 

3497 

3543 

941 

3590 

3636 

3682 

3728 

3774 

3820 

3866 

3913 

3969 

4005 

942 

4051 

4097 

4143 

4189 

4235 

4281 

4327 

4374 

4i20 

4466 

943 

4512 

4558 

4604 

4650 

4696 

4742 

4788 

4834 

4880 

4926 

944 

4972 

5018 

5064 

5110 

5156 

5202 

5248 

5294 

5340 

5386 

945 

5432 

5478 

5524 

5570 

5616 

5662 

5707 

5753 

6799 

5845 

946 

5891 

5937 

5983 

6029 

6075 

6121 

6167 

6212 

6258 

6304 

947 

6350 

6396 

6442 

6488 

6533 

6579 

6925 

6671 

6717 

6763 

948 

6808 

6854 

6900 

6946 

6992 

7037 

7083 

7129 

7176 

7220 

949 

7266 

7312 

7358  j  7403 

7449 

7495 

7541 

7586 

7632 

7678 

19 


20 

LOGARITHMS 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

■ 
9 

950 

977724 

7769 

7815 

7861 

7906 

7952 

7998 

8043 

8089 

8135 

951 

8181 

8226 

8272 

8317 

8363 

8409 

8454 

8500 

8546 

8591 

952 

8637 

8683 

8728 

8774 

8819 

8865 

8911 

895(3 

9002 

9047 

953 

9093 

9138 

9184 

9230 

9275 

9321 

9360 

9412 

9457 

9508 

954 

9548 

9594 

9639 

9685 

9730 

9776 

9821 

9867 

9912 

9958 

955 

980003 

0049 

0094 

0140 

0185 

0231 

0276 

0322 

0367 

0412 

956 

0458 

0503 

0549 

0594 

0640 

0685 

0730 

0776 

0821 

0867 

957 

0912 

0957 

1003 

1048 

1093 

1139 

1184 

1229 

1275 

1320 

958 

1366 

1411 

1466 

1501 

1547 

1592 

1637 

1683 

1728 

1773 

959 

1819 

1864 

1909 

1964 

2000 

2045 

2090 

2135 

2181 

2226 

960 

2271 

2316 

2362 

2407 

2452 

2497 

2543 

2588 

2633 

2678 

961 

2723 

2769 

2814 

2859 

2904 

2949 

2994 

3040 

3085 

3130 

962 

3175 

3220 

3265 

3310 

3356 

3401 

3446 

3491 

3536 

3581 

963 

3626 

3671 

3716 

3762 

3807 

3862 

3897 

3942 

3987 

4032 

964 

4077 

4122 

4167 

4212 

4257 

43  J2 

4347 

4392 

4437 

4482 

965 

4527 

4572 

4617 

4662 

4707 

4752 

4797 

4842 

4887 

4932 

966 

4977 

5022 

5067 

5112 

5157 

6202 

6247 

5292 

5337 

6382 

967 

B426 

6471 

6516 

5661 

6606 

6651 

5699 

6741 

5786 

5830 

968 

6875 

5920 

5965 

6010 

6055 

6100 

6144 

6189 

6234 

6279 

969 

6324 

6369 

6413 

6468 

6503 

6548 

6593 

6637 

6682 

6727 

970 

6772 

6817 

6861 

6906 

6961 

6996 

7040 

7035 

7130 

7176 

971 

7219 

7264 

7309 

7353 

7398 

7443 

7488 

7532 

7577 

7622 

972 

7666 

7711 

7766 

7800 

7846 

7890 

7934 

7979 

8024 

8068 

973 

8113 

8157 

8202 

8247 

8291 

8336 

8381 

8425 

8470 

8514 

974 

8559 

8604 

8648 

8693 

8737 

8782 

8826 

8871 

8916 

8960 

975 

9005 

9049 

9093 

9138 

9183 

9227 

9272 

9316 

9301 

9405 

976 

9450 

9494 

9539 

9583 

9628 

9672 

9717 

9761 

9803 

9850 

977 

9895 

9939 

9983 

..28 

..72 

.117 

.161 

.208 

.250 

.294 

978 

990339 

0383 

0428 

0472 

0516 

0561 

0605 

0650 

0694 

0738 

979 

0783 

0827 

0871 

0916 

0960 

1004 

1049 

1093 

1137 

1182 

980 

1226 

1270 

1315 

1369 

1403 

1448 

1492 

1536 

1580 

1626 

981 

1669 

1713 

1768 

1802 

1846 

1890 

19.'?5 

1979 

2023 

2067 

982 

2111 

2156 

2200 

2244 

2288 

2333 

2377 

2421 

2465 

2509 

983 

2554 

2598 

2642 

2686 

2730 

2774 

2819 

2863 

2907 

2951 

984 

2995 

3039 

3083 

3127 

3172 

3216 

3260 

3304 

3348 

3392 

985 

3436 

3480 

3524 

3668 

3613 

3657 

3701 

3745 

3789 

3833 

986 

3877 

3921 

3966 

4009 

4053 

4097 

4141 

4185 

4229 

4273 

987 

4317 

4361 

4405 

4449 

4493 

4537 

4581 

4625 

4669 

4713 

988 

4757 

4801 

4845 

4886 

4933 

4977 

5021 

6066 

5108 

5152 

989 

5196 

5240 

6284 

6328 

5372 

6416 

5460 

6504 

5647 

5691 

990 

5635 

5679 

5723 

5767 

5811 

5864 

6898 

5942 

5986 

^30 

991 

6074 

6117 

6161 

6205 

6249 

6293 

6337 

6380 

6424 

6468 

992 

6512 

6555 

6599 

6643 

6687 

6731 

6774 

6818 

6862 

6906 

993 

6949 

6993 

7037 

7080 

7124 

7168 

7212 

7255 

7299 

7343 

994 

7386 

7430 

7474 

7617 

7661 

7605 

7648 

7692 

7736 

7779 

995 

7823 

7867 

7910 

7954 

7998 

8041 

8086 

8129 

8172 

8216 

996 

8259 

8303 

8347 

8390 

8434 

8477 

8521 

8564 

8608 

8652 

997 

8695 

8739 

8792 

8826 

8869 

8&13 

8956 

9000 

9043 

9087 

998 

9131 

9174 

9218 

9261 

9306 

9348 

9392 

9435 

9479 

9522 

999 

9565 

9609 

9652 

9696 

9739 

9783 

9826 

9870 

9913 

9957 

TABLE  ir.    Log.  Sines  and  Tangents.  (0°)  Natural  Sines.         21 

/ 

Sine. 

D.IO" 

Cosine. 

D.IO" 

Tang. 

D.IO" 

Coiang. 

N.sine. 

N.  COS. 

0 

0.000000 

10.000000 

0.000000 

Infinite. 

00000 

100000 

60 

1 

6.463726 

000000 

6.463726 

18.536274 

00029 

100000 

59 

2 

764756 

000000 

764756 

235244 

00058 

100000 

58 

3 

940847 

000000 

940847 

059153 

00087 

100000 

57 

4 

7.065786 

000000 

7.065786 

12.934214 

00116 

100000 

56 

5 

162696 

000000 

162696 

837304 

00145 

100000 

55 

6 

241877 

9.999999 

241878 

758122 

00175 

100000 

54 

7 

308824 

999999 

308825 

691175 

00204 

lOOODO 

53 

8 

366816 

999999 

366817 

633183 

00233 

100000 

52 

9 

417968 

999999 

417970 

582030 

00262 

100000 

51 

10 

463725 

999998 

463727 

536273 

00291 

100000 

50 

11 

7.505118 

9.999998 

7.505120 

12.494880 

003201  99999 

49 

12 

542908 

999997 

542909 

457091 

00349 

99999 

48 

13 

577668 

999997 

577672 

422328 

00378 

99999 

47 

14 

609853 

999996 

609857 

390143 

00407 

99999 

46 

15 

639816 

999996 

639820 

360180 

00436 

99999 

46 

16 

667845 

999995 

667849 

332151 

00465 

99999 

44 

17 

694173 

999995 

694179 

305821 

00495 

99999 

43 

18 

718997 

999994 

719003 

280997 

00524 

99999 

42 

19 

742477 

999993 

742484 

257516 

00553 

99998 

41 

20 

764754 

999993 

764761 

235239 

00582 

99998 

40 

21 

7.785943 

9.999992 

7.785951 

12.214049 

00611 

99998 

39 

22 

806146 

999991 

806155 

193845 

00640 

99998 

38 

23 

825451 

999990 

825460 

174540 

00669 

99998 

37 

24 

843934 

999989 

843944 

156056 

00698 

99998 

36 

25 

861663 

999988 

861674 

138326 

00727 

99997 

35 

26 

878695 

999988 

878708 

121292 

00756 

99997 

34 

27 

895085 

999987 

895099 

104901 

00785 

99997 

33 

28 

910879 

999986 

910894 

089106 

00814 

99997 

32 

29 

928119 

999985 

926134 

073866 

00844 

99996 

31 

30 

940842 

999983 

940858 

059142 

00873 

99996 

30 

31 

7.955082 

2298 
2227 
2161 
2098 
2039 
1983 

9.999982 

0.2 

0.2 

7.955100 

2298 
2227 
2161 
2098 
2039 
1983 
1930 
1880 
1833 
1787 
1744 
1703 
1664 
1627 
1591 
1557 
1524 
1493 
1463 
1434 
1406 
1379 
1353 
1328 
1304 
1281 
1259 
1238 
1217 

12.044900 

00902 

99996 

29 

32 

968870 

999981 

968889 

031111 

00931 

99996 

28 

33 

'  982233 

999980 

982253 

017747 

00960 

99995 

27 

34 

995198 

999979 

0*2 

995219 

004781 

00989 

99995 

26 

35 

8.007787 

999977 

0-2 
0-2 

8.007809 

11.992191 

01018 

99995 

26 

36 

020021 

999976 

020045 

979955 

01047 

99995 

24 

37 

031919 

999975 

0-2 

031945 

968055 

01076 

99994 

23 

38 

043501 

1930 
1880 
1832 
1787 
1744 
1703 
1664 
1626 
1591 
1557 
1524 
1492 
1462 
1433 
1405 
1379 
1353 
1328 
1304 
1281 
1259 
1237 
1216 

999973 

0-2 
0-2 

0-2 
0-2 
0-2 
0*2 
0'2 
0*3 
0'3 

o;3 

0.3 
0.3 
0.3 
0.3 
0.3 
0.3 
0.3 
0.3 
0.3 
0.4 
0.4 
0.4 
0.4 

043527 

956473 

01105 

99994 

22 

39 

054781 

999972 

054809 

945191 

01134 

99994 

21 

40 

085776 

999971 

065806 

934194 

01164 

99993 

20 

41 

8.076500 

9.999969 

8.076531 

11.923469 

01193 

99993 

19 

42 

086965 

999968 

086997 

913003 

01222 

99993 

18 

43 

097183 

999966 

097217 

902783 

01251 

99992 

17 

44 

107167 

999964 

107202 

892797 

01280 

99992 

16 

45 

116926 

999963 

116963 

883037 

01309 

99991 

15 

46 

126471 

999961 

126610 

873490 

01338 

99991 

14 

47 

135810 

999959 

135851 

864149 

01367 

99991 

13 

48 

144953 

999958 

144996 

855004 

01396 

99990 

12 

49 

153907 

999956 

153952 

846048 

01425 

99990 

11 

50 

162681 

999954 

162727 

837273 

01454 

99989 

10 

51 

8.171280 

9.999952 

8.171328 

11.828672 

01483 

99989 

9 

52 

179713 

999950 

179763 

820237 

01513 

99989 

8 

53 

187985 

999948 

188036 

811964 

01542 

99988 

7 

54 

196102 

999946 

196156 

803844 

01571 

99988 

6 

55 

204070 

999944 

204126 

795874 

01600 

99987 

6 

56 

2X1895 

999942 

211953 

788047 
780359 

01629 

99987 

4 

57 

219581 

999940 

219641 

01658 

99986 

3 

58 

227134 

999938 

227195 

772805  i 

01687 

99986 

2 

59 

234557 

999936 

234621 

765379 ! 

01716 

99985 

1 

60 

241855 

999934 

241921 

758079 ; 

01745 

99985 

0 

/ 

Cosine. 

Sine. 

Co  til  lie:. 

'I'anff.  ' 

N.  COS. 

N.  sine- 

89  Degrees.                           | 

22 


Log.  Sines  and  Tangents.    (1°)     Natural  Sines,       TABLE  II. 


Sine. 


0  8.241855 

1  9.dQf«5i 


1   249033 
256094 

3  263042 

4  "'^"'^- 
5 
6 
7 
8 
9 

10 

118 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21  8 

22 

23 

24 

25 

26 

27 


1196 

1177 

1158 

1140 

1122 

1105 

1088 

1072 

1056 

- -:  1041 

o^^  1027 

''^"   1012 


276614 
283243 
289773 
296207 
302546 
308794 
.314954 
321027 
327016 
332924 
338763 
344504 
350181 


985 
971 
959 
946 

355783  322 
361315  qi/j 
366777 
.372171 

377499  ^;; 

382762  Q^i 

387962  ^L 

393101  g^ 

398179  Q^ 

403199  ^i 

408161  gio 

413068  SAq 

417919  S^n 

8.422717  ^XV 

I  427462  'l^ 

432156  ^«f 

436800  7fi^ 

441394  l^ 

446941  'Tq 

460440  742 

454893  «oK 

459301  70? 

40  463666  LiL 

41  8.467986  if^ 

42  472263  XA^ 

43  476498  '^ 

44  480693  ^^ 
46  484848  ^^^ 

46  488963!  ^^^ 

47  4930401  ^'^ 

48  4970781  XA« 

49  601080  X^; 
60  606045  1  ^J 
51  8.608974  ^^J 
62  512867  ^J^ 

516726  ^^^ 

520651  ^^' 

624343  ?.t>i 

628102  ^^^ 

631828  ^;^ 

635523  ^Ji 

539186  ^^^ 

642819  ^^^ 


29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 


D.IO"  Cosine. 


53 
54 
55 
56 
57 
68 
59 


9.999934 
999932 
999929 
999927 
999925 
999922 
999920 
999918 
999915 
999913 
999910 

9.999907 
999905 
999902 
999899 
999897 
999894 
999891 
999888 


I  Cosir 


D.IO' 


9.999879 
999876 
999873 
999870 
999867 
999864 
999861 
999858 
999854 
999861 

9.999848 
999844 
999841 
999838 
999834 
999831 
999827 
999823 
999820 
999816 

9.999812 
999809 
999805 
999801 
999797 
999793 
999790 
999786 
999782 
999778 

9.99977.f 
999769 
999765 
999761 
999757 
999753 
999748 
999744 
999740 
999735 


0-4 
0.4 
0.5 
0.5 
0.5 
0.5 
06 
0.5 
0.5 
0.6 
0.6 
0.5 
0.5 
0.5 
0.5 
0.5 
0.5 
0.5 
0-6 
0.6 
0.6 


Sine. 


0.6 
0.6 
0-6 
0.6 
0.6 
0-6 
0-6 
0.6 
0.6 
0.7 
0-7 
0.7 
0.7 
0.7 
0.7 
0.7 
0.7 
0.7 
0.7 
0.7 
0.7 
0.7 
0.7 
0.7 


8.241921 
249102 
256165 
263116 
269956 
276691 
283323 
289856 
296292 
302634 
308884 

8.315046 
321122 
327114 
333025 
333856 
344610 
350289 
355895 
361430 
366895 

8.372292 
377622 


Tang. 


388092 
393234 
398315 
403338 
408304 
413213 
418068 

8.422869 
427618 
432315 
436962 
441560 
446110 
450613 
455070 
459481 
463849 

8.468172 
472454 
476693 
480892 
485060 
489170 
493250 
497293 
501298 
505267 
509200 
513098 
5169ol 
620790 
524586 
528349 
532080 
535779 
539447 
543084 


Cnlaii^r- 


D.IO' 


1197 
1177 
1158 
1140 
1122 
1105 
1089 
1073 
1057 
1042 
1027 
1013 
999 
985 
972 
959 
946 
934 
922 
911 
899 
888 
879 
867 
857 
847 
837 
828 
818 
809 
800 
791 
783 
774 
766 
758 
750 
743 
735 
728 
720 
713 
707 
700 
693 
686 
680 
674 
668 
661 
656 
650 
644 
638 
633 
627 
622 
616 
611 
606 


Cotang. 


11.758079 
750898 
743835 
736885 
730044 
723309 
716677 
710144 
703708 
697366 
691118 

11-684954 
678878 
672886 
666975 
661144, 
055390 
649711 
644105 
638570 
633105 

11.627708 
622378 
617111 
611908 
606766 

■  601685 
596662 
691696 
586787 
581932 

11.577131 
572382 
667685 
563038 
558440 
553890 
549387 
644930 
540519 
536161 

11 .531828 
527546 
523307 
619108 
614950 
510830 
506750 
502707 
498702 
494733 

11.490800 
486902 
483039 
479210 
475414 
471651 
467920 
464221 
460653 
456916 


N.  sine.  N.  cos. 


01742 
01774 
01803 
01832 
01862 
01891 
01920 
i  01949 
01978 
02007 
02U31 
02065 
02094 
0212s 

miou 

02181 

02211 

02240 

02269 

02298 

02327 

02356 

02385 

02414 

02443 

024' 

02501 

02530 

02560 


99985 
99984 
99934 
99983 
99983 
99982 
99982 
99^81 
99980 
99980 
^9979 
)y979 
99978 
99977 
J:.977 
99976 
99976 
i>9975 
Jy974 
99974 
99973 
yi>972 
99972 
99971 
99970 
2  y9969 
99969 
99968 
99967 


02589  99966 


'i"^ 


02618 
02647 
02076 
02705 
02734 
102763 
02792 
02821 
02850 
02879 
02908 
02938 
02967 
0299G 
03026 
03054 
03083 
03112 
03141 
03170 
03199 
03228 
0q257 

I  03286 
103316 
103645 

I I  0337 
!  103403 
1:03432 
j(  03461 
I ! 03490 
I '  N.  COS. 


^9966 
99965 
99964 
99963 
99963 
99962 
99961 
99960 
99959 
99959 
99958 
99957 
99956 
99955 
99954 
99953 
99952 
99952 
99951 
99950 
99949 
99948 
99947 
99946 
99945 
99944 
4,99943 
99942 
99941 
99940 
y9939 


88  Degreep. 


TABLE  II.        Log.  Sines  and  Tangents.    (-P)    Natural  Sines. 


23 


iS.ne.   D.  10"  Cosine.  D.  10"|  Tang.   D.  10"  Colang.  j jN.  sine.  N.  cos 


8.542819 
54G422 
549995 
553539 
557054 
560540 
663999 
567431 
570836 
574214 
677566 

8.580892 
684193 
587469 
590721 
593948 
597152 
600332 
603489 
606623 
609734 

8.612823 
615891 
618937 
621962 
624965 
627948 
630911 
633854 
636776 
639680 

8.642663 
645428 
648274 
651102 
65.3911 
656702 
659475 
662230 
664968 
667689 
670393 
673080 
675751 
678405 
681043 
683665 
686272 
688863 
691438 
693998 

8.696543 
699073 
701589 
704090 
706577 
709049 
711507 
713952 
716383 
718800 


Cosine. 


600 
595 

591 
586 
581 
576 
672 
667 
563 
559 
654 
550 
646 
642 
538 
634 
530 
526 
622 
619 
615 
511 
608 
504 
501 
497 
494 
490 
487 
484 
481 
477 
474 
471 
468 
466 
462 
459 
466 
453 
451 
448 
445 
442 
440 
437 
434 
432 
429 
427 
424 
422 
419 
417 
414 
412 
410 
407 
405 
403 


,999735 
999731 
999726 
999722 
999717 
999713 
999708 
999704 
999G99 
999694 
999689 
.999685 
999680 
999675 
999670 
999665 
999660 
999665 
999650 
999645 
999640 
.999635 
999629 
999324 
999619 
999614 
999608 
999603 
999597 
999592 
999586 
.999581 
999575 
999570 
999564 
999558 
999553 
999547 
999641 
999535 
999629 
.999624 
999518 
999512 
999506 
999500 
999493 


999481 
999475 
999469 
.999463 
999456 
999450 
999443 
999437 
999431 
999424 
999418 
999411 
999404 


Si  I 


0.7 

0.7 

0.7 

0-8 

0-8 

0-8 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

0.9 

0.9 

0.9 

0.9 

0.9 

09 

0.9 

0.9 

0.9 

0.9 

0-9 

0.9 

0.9 

0.9 

0.9 

1.0 

1-0 

1.0 

1.0 

1-0 

1.0 

1-0 

1.0 

1-0 

1-0 

1.0 

1.0 

1.0 

1.0 

1.0 

1.0 

1 

1 

1 


.643084 
546691 
650268 
553817 
557336 
560828 
564291 
567727 
571137 
574520 
577877 
.581208 
584614 
587795 
691051 
594283 
597492 
600677 
603839 
606978 
610094 
.613189 
616262 
619313 
622343 
625352 
628340 
631308 
634256 
637184 
640093 
.642982 
646863 
648704 
651537 
654362 
657149 
659928 
662689 
665433 
668160 
.670870 
673663 
676239 
678900 
681544 
684172 
6  6784 
689381 
691963 
694529 
.697081 
699617 
702139 
704246 
707140 
709618 
702083 
714634 
716972 
719396 


Cotang. 


602 

596 
591 
587 
682 
677 
573 
668 
664 
659 
555 
551 
547 
543 
539 
535 
531 
527 
523 
519 
516 
512 
508 
605 
501 
498 
495 
491 
488 
485 
482 
478 
475 
472 
469 
466 
463 
460 
457 
454 
453 
449 
446 
443 
442 
438 
435 
433 
430 
428 
425 
423 
420 
418 
415 
413 
411 
408 
406 
404 


11.456916!  I  03490 

453309  1 1  03519 

4497321103548 

446183;  0357 

442664; 

439172: 

435709  1 1 

432273  1' 

428863  I 

425480 ; 

422123  I 
11.418792, 

415486 

412205 

408949 

405717 

402508 

399323 

396161 

393022 

389906 
11.386811 

383738 

380687 


03606 
03635 
03664 
03693 
03723 
03752 
03781 
03810 
03839 
03868 
03897 
03926 
03955 
03984 
04013 
04042 
04071 
04100 
03129 
04159 


377657  104188 
374648!  04217 
371660:104246 
04275 
04304 
04333 
04362 
04391 
04420 
04449 
04478 
04507 
04536 
04 

04594 
04623 
04653 
04682 
04711 
04740 
04769 
04798 
04827 


368692'! 
365744 ' 
362816 
359907 
,357018 
354147 
351296 
348463 
346648 
342851 
340072 
337311 
334567 
331840 
11.329130 
326437 
323761 
321100 
318456 
316828 

313216  1 104856 
310619!  104885 
308037  1104914 
305471 
302919 
300383 
297861 
295354 
292860 
290382 
287917 
285465 
283028 
280604 


04943 
04972 
05001 
05030 
05059 
0508b 
051 1: 
U5146 
05175 
05205 
05234 


Tanff.   !  N.  COS.  N.sine. 


99939 
99938 
99937 
99936 
99935 
99934 
99933 
99932 
99931 
99930 
99929 
99927 
99926 
99925 
99924 
99923 
99922 
99921 
99919 
99918 
99917 
99916 
99915 
99913 
99912 
99911 
99910 
99909 
99907 
99906 
99905 
99904 
99902 
99901 
99900 
99898 
99897 


05-99896 
99894 
^9893 
99892 
99890 
99889 
99888 
99886 
99885 
99883 
99882 
99881 
99879 
99878 
99876 
99875 
99873 
998  ?2 
99870 
99869 
9^867 
99866 
99864 
99S6o 


60 
59 
58 
57 
66 
56 
64 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 


87  Degrees. 


24 


Log,  Sines  and  Tangents.     (3°;    Natural  Sines.        TABLE  II. 


0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


Sine. 


8.718800 
721204 
723595 
725972 
728337 
730688 
733027 
735354 
737667 
739969 
742259 

8.744536 
746802 
749055 
751297 
753528 
755747 
757955 
760151 
762337 
764511 

8.766675 
768828 
770970 
773101 
775223 
777333 
779434 
781524 
783605 
785675 

8.787736 
789787 
791828 
793859 
795881 
797894 
799897 
801892 
803876 
805852 

8.807819 
809777 
811726 
813667 
815599 
817522 
819436 
821343 
823240 
825130 

8.827011 
828884 
830749 
832607 
834456 
836297 
838130 
839956 
841774 
843585 


Cosine. 


D.  10"     Cosine. 


401 
398 
396 
394 
392 
390 
388 
386 
384 
382 
380 
378 
376 
374 
372 
370 


364 
362 
361 
359 
357 
355 
353 
352 
350 
348 
347 
345 
343 
342 
340 
339 
337 
335 
334 
332 
331 
329 
328 
326 
325 
323 
322 
320 
319 
318 
316 
315 
313 
312 
311 
309 
308 
307 
306 
304 
303 
302 


1.999404 
999398 
999391 
999384 
999378 
999371 
999364 
999357 
999350 
999343 
999336 

1.999329 
999322 
999315 
999308 
999301 
999294 
999286 
999279 
999272 
999265 

1.999257 
999250 
999242 
999235 
999227 
999220 
999212 
999205 
999197 
999189 

.999181 
999174 
999166 
999158 
999150 
999142 
999134 
999126 
999118 
999110 

.999102 
999094 
999086 
999077 
999069 
999061 
999053 
999044 
999036 
999027 

.999019 
999010 
999002 
998993 
998984 
998976 
998967 
998958 
998950 
998941 


Sine. 


D.  10' 


1. 

1. 

1. 

1. 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.3 

1.3 

1.3 

1.3 

1.3 

1.3 

1.3 

1.3 

1.3 

1.3 

1.3 

1.3 

1.3 

1.3 

1.3 

1.3 

1.3 

1.3 

1.3 

1.3 

1.4 

1.4 

1.4 

1.4 

1.4 

1.4 

1.4 

1.4 

1.4 

1.4 

1.4 

1.4 

1.4 

1.4 

1.4 

1.5 

1.5 

1.5 


Tang. 


D.  10' 


8.719396 
721806 
724204 
726588 
728959 
731317 
733663 
735996 
738317 
740626 
742922 

8.745207 
747479 
749740 
751989 
754227 
756453 
758668 
760872 
763065 
765246 

8.767417 
769578 
771727 
773866 
775995 
778114 
780222 
782320 
784408 
786486 

8.788554 
790613 
792662 
794701 
796731 
798752 
800763 
802765 
804858 
806742 

8.808717 
810683 
812641 
814589 
816629 
818461 
820384 
822298 
824205 
826103 

8.827992 
829874 
831748 
833613 
835471 
837321 
839163 
840998 
842825 
844644 


Cotanjr. 


402 
399 
397 
395 
393 
391 
389 
387 
385 
383 
381 
379 
377 
375 
373 
371 
369 
367 
365 
364 
362 
360 
358 
356 
355 
353 
351 
350 
348 
346 
345 
343 
341 
340 
338 
337 
335 
334 
332 
331 
329 
328 
326 
325 
323 
322 
320 
319 
318 
316 
315 
314 
312 
311 
310 
308 
307 
306 
304 
303 


05234 
I  05263 
] 05292 
05321 
I  05350 
i  05379 
05408 
05437 
I  05466 
; 05495 
05524 
05553 
05582 
05611 
05640 
05669 
05698 
05727 
05756 
05785 
05814 
05844 
05873 
05902 
05931 


Cotang.  |fN.  sine 

11.280604 

278194 

275796 

273412 

271041 

268683 

266337 

264004 

261683 

259374 

257078 
11 .254793 

252521 

250260 

248011 

245773 

243547 

241332 

239128 

236935 

234754 
11.232583 

230422 

228273 

226134 

224005 

221886 

219778 

217680 

215592 

213514 
11.211446 

209387 

207338 

205299 

203269 

201248 

199237 

197235 

195242 

193258 
11.191283 

189317 

187359 

185411 

183471 

181539 

179616 

177702 

175795 

173897 
11.172008 

170126 

168252 

166387 

164529 

162679 

160837 

159002 

157175 

155356 


99863 

99861 

99860 

99858 

9985 

99856 

99854 

99852 

99851 

99849 

99847 

99846 

99844 

99842 

99841 

99839 

99838 

99836 

99834 

99833 

99831 

99829 

9982 

99826 

99824 


0596099822 


05989 
06018 
06047 

I  06076 
06105 
06134 
06163 

1106192 

!  106221 
06250 
06279 
06308 
06337 
06366 
06395 
06424 
06453 
06482 
06511 
06540 
01)569 
06598 
06627 
06656 

,  06685 

I  06714 
:!  06743 
;!  06773 
;i  06802 
;'  06831 

I I  06860 
1 06889 
108918 
106947 

':  06976 


Tang. 


N.  COS. 


99821 

99819 

9981 

99815 

99813 

99812 

99810 

99808 

99806 

99804 

99803 

99801 

99799 

99797 

99795 

99793 

99792 

99790 

99788 

99786 

99784 

99782 

99780 

99778 

99776 

99774 

99772 

99770 

99768 

99766 

99764 

99762 

99760 

99758 

99756 


N.  COS.  N.sine, 


86  Degrees. 


TABLE  II.        Log.  Sines  and  Tangents.     (4°)    Natural  Sines. 


25 


Sine.      D.  10" 


Cosine. 


Cosine. 


.998941 
998932 
998923 
998914 
998905 


998887 
998878 
998869 
998860 
998861 
.998841 
998832 
998823 
998813 
998804 
998795 
998785 
998776 
998766 
998757 
.998747 
998738 
998728 
998718 
998708 


998679 
998669 
998669 
.998649 
998639 
998629 
998619 
998609 
998599 
998589 
998678 
998568 
998558 
.998548 
998537 
998527 
998516 
998506 
998495 
998486 
998474 
998464 
998463 
.998442 
998431 
998421 
998410 
998399 
r98388 
998377 
998366 
998355 
998344 


Sine. 


D.  10" 


Tang. 

.844644 
846455 
848260 
850057 
851846 
853628 
855403 
867171 
858932 
860686 
862433 

.864173 
865906 
867632 
869361 
871064 
872770 
874469 
876162 
877849 
879529 

.881202 
882869 
884530 
886185 
887833 
889476 
891112 
892742 
894366 
895984 

;.  897596 
899203 
900803 
902398 
903987 
905570 
907147 
908719 
910285 
911846 

;.  913401 
914951 
916495 
918034 
919568 
921096 
922619 
924136 
925649 
927156 

!.  928658 
930155 
931647 
933134 
934616 
936093 
937565 
939032 
940494 
941952 
Cotang. 


D.  10" 


302 
301 

299 
298 
297 
296 
295 
293 
292 
291 
290 
289 
288 
287 
285 
284 
283 
282 
281 
280 
279 
278 
277 
276 
275 
274 
273 
272 
271 
270 
269 
268 
267 
266 
265 
264 
263 
262 
261 
260 
259 
258 
257 
256 
256 
255 
254 
253 
252 
251 
250 
249 
24y 
248 
247 
246 
245 
244 
244 
243 


Cotang.  I  IN.  sine.  N.  cos. 


11 


11.155356 
153545 
151740 
149943 
148154 
146372 
144597 
142829 
141068 
139314 
137567 
135827 
134094 
132368 
130649 
128936 
127230 
125531 
123838 
122151 
120471 

11.118798 
117131 
115470 
113815 
112167 
110524 
108888 
107258 
105634 
104016 

11.102404 
100797 
099197 
097602 
096013 
094430 
092853 
091281 
089715 
088154 

11.086599 
085049 
083505 
081966 
080432 
078904 
077381 
075864 
074351 
072844 
071342 
069845 
068353 
066866 
065384 
063907 
062435 
060968 
059506 
058048 


06976 
07005 
07034 
07063 
07092 
07121 
07150 
07179 
07208 
07237 
07266 
07295 
07324 
07353 
07382 
07411 
07440 
07469 
07498 
07627 
07556 
07585 


07643 
07672 
07701 
07730 
I  j  07759 
I!  077 
li  07817 
!j  07846 
II 07875 
■  I  07904 
I  07933 
^07962 
!  I  07991 
i 108020 
108049 
i  08078 
I  08107 


99756 
99754 
99752 
99760 
99748 
99746 
99744 
99742 
99740 
99738 
99736 
99734 
99731 
99729 
99727 
99725 
99723 
99721 
99719 
99716 
99714 
99712 


0761499710 


99708 
99705 
99703 
99701 
99699 
99696 
99694 
99692 
99689 
99687 
99685 
99683 
99680 
99678 
99676 
99673 
y9671 


08136  99668 


11 


108165 
108194 
108223 
j I  08252 
1 1 08281 
i]  08310 
1 108339 
'!  08368 
I  j 08397 
i  08426 
I  j  08455 
i 1 08484 
108513 
: 08542 
-085 
I  i 08600 
'108629 
:  108658 

ijose 

: 08716 


N.  COS.  N.sine. 


99666 
99664 
99661 
99659 
99657 
99654 
99652 
99649 
99647 
99644 
99642 
99639 
99637 
99635 
99632 
99630 
9y627 
99625 
99622 
99619 


60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
46 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
33 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 


85  Degrees. 


26 


Log.  Sines  and  Tangents.    (5°)     Natural  Sines.        TABLE  11, 


2 
3 

4 
B 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 


Sine. 

8.940296 
941738 
943174 
944608 
946034 
947456 
948874 
950287 
951696 
953100 
954499 

8.955894 
957284 
958670 
960052 
961429 
962801 
964170 
965534 
966893 
968249 

8.969600 
970947 
972289 
973628 
974962 
976293 
977619 
978941 
980259 
981573 

8.982883 
984189 
985491 
986789 
988083 
989374 
990660 
991943 
993222 
994497 

8.995768 
997036 
998299 
999560 

[9.000816 
002089 
003318 
004563 
005805 
007044 

9.008278 
009510 
010737 
011962 
013182 
014400 
015613 
016824 
018031 
019235 


D.  10 


240 

239 

239 

238 

237 

236 

235 

235 

234 

233 

232 

232 

231 

230 

229 

229 

228 

227 

227 

226 

225 

224 

224 

223 

222 

222 

221 

220 

220 

219 

218 

218 

217 

216 

216 

215 

214 

214 

213 

212 

212 

211 

211 

210 

209 

209 

208 

208 

207 

206 

206 

205 

205 

204 

203 

203 

202 

202 

201 

201 


Cosine. 


9.998344 
998333 
998322 
998311 
998300 
998289 
998277 
998266 
998255 
998243 
998232 
998220 
998209 
998197 
998186 
998174 
998163 
998151 
998139 
998128 
998116 

9.998104 
998092 
998080 
998068 
998056 
998044 
998032 
998020 


997996 
9.997984 
997972 
997959 
997947 
997935 
997922 
997910 
997897 
997885 
997872 
9.997860 
997847 
997835 
997822 
997809 
997797 
997784 
997771 
997758 
997745 
997732 
997719 
997706 
997693 
997680 
997667 
997654 
997641 
997628 
997614 


Cosine. 


D.  10" 


1.9 
1.9 
1.9 
1.9 
1.9 
1.9 
1.9 
1.9 
1.9 
1.9 
1.9 
1.9 
1.9 
1.9 
1.9 
1.9 
1.9 
1.9 
2.0 
2.0 
2.0 
2.0 
2.0 
2.0 
2.0 
2.0 
2.0 
2.0 
2.0 
2.0 
2.0 
2.0 
2.0 


Tang. 


2.1 
2.1 
2.1 
2.1 
2.1 
2.1 
2.1 
2.1 
2.1 
2.1 
2.1 
2.1 
2.1 
2.1 
2.1 
2.1 
2.2 
2.2 
2.2 
2.2 
2.2 
2.2 


Sine. 


8.941952 
943404 
944852 
946295 
947734 
949168 
950597 
952021 
953441 
954856 
956267 

8.957674 
959075 
960473 
961866 
963255 
964639 
966019 
967394 
968766 
970133 

8.971496 
972855 
974209 
975560 
976906 
978248 
979586 
980921 
982251 
983577 

8.984899 
986217 
987532 
988842 
990149 
991461 
992750 
994045 
995337 
996624 

8.997908 
999188 
000465 
001738 
003007 
004272 
005534 
006792 
008047 
009298 
010546 
011790 
013031 
014268 
015502 
016732 
017959 
019183 
020403 
021620 


D.  10"l  Cotang.  {iN.  sine.  N.  cos 


Cotang. 


242 

241 

240 

240 

239 

238 

237 

237 

236 

235 

234 

234 

233 

232 

231 

231 

230 

229 

229 

228 

227 

226 

226 

225 

224 

224 

223 

222 

222 

221 

220 

220 

219 

218 

218 

217 

216 

216 

215 

215 

214 

213 

213 

212 

211 

211 

210 

210 

209 

208 

208 

207 

207 

206 

206 

205 

204 

204 

203 

203 


11.058048! '08716 
056596  i  I  08745 
0551481108774 
0537051108803 
052266!  1 08831 
050832  1 08860 
049403 
047979!  108918 
046559!  108947 
045144 'i  08976 
043733 

11.042326 
040925 
039527 
038134 
036745 
035361 
033981 
032606 
031234 
029867 

11.028504 
027145 
025791 


'■  09005 

■  09034 

' 09063 

1 09092 

109121 

109150 

i 09179 

1 09208 

109237 

! 09266 

1 09295199567 

I  09324  99564 

'  09353199562 

:  09382!99559 
024440  j  I  09411 199556 
023094  :l09440i99553 
021752  1 109469199551 
0204141 1 09498199548 
0190791 1  09527199545 


99619 

99617 

99614 

99612 

99609 

9960 

99604 

99602 

99599 

99596 

99594 

99591 

99588 

99586 

99583 

99580 

99578 

99575 

99572 

99570 


60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
46 
44 
43 
42 
41 
40 
39 
38 
37 
36 
36 
34 
33 
32 


017749 
016423 
.015101 
013783 
012468 
011158 
009851 


09556 1995421  31 


09585199540 
09614J99537 
09642 199534 
09671,99531 
09700199528 
09729199526 


008549  ii06758|99523 
007250  l|  09787  99520 
005955  I  j  09816199517 
004663  11 09845199514 
003376  1  09874199511 


.002092 
000812 
10.999535 
998262 
996993 
995728 


09903  ;99508 
09932  J99o0t 
09961199503 
09990199500 
10019:99497 
10048|99494 
994466!!  10077199491 
9932081110106  99488 
991953!  1 10135|99485 
990702  i:  1016499482 
10192,99479 
1022199476 
10250  99473 
10279  99470 
10308 199467 
983268  I i  1033/199464 
983041  1 1 10366,99461 
980817;!  10395199458 
979597  1 1 10424199155 
978380  ij  10453  J99452 
Tang.   I !  N.  cos.|N.pine. 


10.989454 
988210 
686969 
985732 
984498 


84  Degrees. 


TABLE  II. 


Log.  Sines  and  Tangents. 


Natural  Sines. 


C!otang.     |  N.  sine.  N.  cos. 


Sine.      D.  10" 


9.019235 
020436 
021632 
022825 
024016 
025203 
026386 
027567 
028744 
029918 
031089 

9.032257 
033421 
034582 
035741 
030896 
038048 
039197 
040342 
041485 
042625 

9.043762 
044895 
046026 
047154 
048279 
049400 
050519 
051635 
052749 
053859 

9.054966 
056071 
057172 
058271 
059367 
060460 
061551 
062639 
063724 
064806 

9.066886 
066962 
068036 
069107 
070176 
071242 
072306 
073366 
074424 
075480 

9.076633 
077683 
078631 
079676 
080719 
081759 
082797 
083832 
084864 
085894 


Cosine. 


200 
199 
199 
198 
198 
197 
197 
196 
196 
195 
195 
194 
194 
193 
192 
192 
191 
191 
190 
190 
189 
189 
180 
188 
187 
187 
186 
186 
185 
185 
184 
184 
184 
183 
183 
182 
182 
181 
181 
180 
180 
179 
179 
179 
178 
178 
177 
177 
176 
176 
176 
176 
176 
174 
174 
173 
173 
172 
172 
172 


Cosine.  D.  10" 


.997614! 
997601 i 
997588  j 
997574 ! 
997561  I 
997547  j 
997534 
997520 ! 
997507  I 
997493  I 
997480  i 

.997466! 
997452 
997439 ! 
997425  i 
997411 
997397 
997383 
997369 
997355 
997341 

.997327 
997313 
997299 
997285 
997271 
997257 
997242 
997228 
997214 
997199 

.997186 
997170 
997156 
997141 
997127 
997112 
997098 
997083 
997068 
997053 

.997039 
997024 
997009 
996994 
996979 
996964 
996949 
996934 
996919 
996904 

.996889 
996874 
996858 
996843 
996828 
996812 
996797 
996782 
996766 
996751 


Sine. 


2.2 

2.2 

2.2 

2.2 

2.2 

2.2 

2.3 

2.3 

2.3 

2 

2 

2 

2 

2. 

2 

2.3 

2-3 

2-3 

2-3 

2-3 

2.3 

2-4 

2-4 

2.4 

2.4 

2-4 

2.4 

2.4 

2.4 


2 

2 

2 

2. 

2 

2 

2.4 

2.4 

2.4 

2.5 

2.5 

2-5 

2-6 

2-5 

2.6 

2.6 

2.6 

2-5 

2.5 

2.5 

2.5 

2.5 

2.6 

2.6 

2.6 

2.6 

2.5 

2.6 

2  6 

2.6 

2.6 


Tang.   iD.  10" 


9.021620 
022834 
024044 
025251 
026455 
027655 
028852 
030046 
031237 
032425 
033609 

9.034791 
035969 
037144 
038316 
039485 
040651 
041813 
042973 
044130 
045284 

9.046434 
047582 
048727 
049869 
061008 
052144 
053277 
054407 
055636 
066659 

9.057781 
058900 
060016 
061130 
062240 
063348 
064453 
065666 
066655 
067762 

9.068846 
069038 
071027 
072113 
073197 
074278 
076356 
076432 
077505 
078576 

9.079644 
080710 
081773 
082833 
083891 
084947 
086000 
087050 
088098 
089144 


Cotang. 
>  Degrees. 


202 
202 
201 
201 
200 
199 
199 
198 
198 
197 
197 
196 
196 
195 
195 
194 
194 
193 
193 
192 
192 
191 
191 
190 
190 
189 
189 
188 
188 
187 
187 
186 
186 
185 
185 
185 
184 
184 
183 
183 
182 
182 
181 
181 
181 
180 
180 
179 
179 
178 
178 
178 
177 
177 
176 
176 
176 
175 
175 
174 


10.9783801 
977166  ; 
975956 ' 
974749  i 
973545 
972345 
971148 
969954 ! 
968763 [ 
967676  i 
966391 [ 

10.965209! 
964031 
962856  i 
961684 1 
960515  I 
959349  I 
958187  I 
957027 ! 
955870 1 
964716  I 

10.953566 
952418 
951273 
950131 
948992 
947856 
946723 
945693 
944465 
943341 

10.942219 
941100 
939984  i 
938870 1 
937760  i 
936652  I 
935547  I 
934444  I 
933346  I 
932248 ! 

10.931164 
930062  I 
928973 
927887 
926803 
925722 
924644 
923568 
922495 
921424 

10.920356 
919290 
918227 
917167 
916109 
915063 
914000 
912960 
911902 
910856 

I   Ta^g^ 


99452 
99449 
99446 
99443 
99440 
99437 
99434 
99431 
9y428 
99424 
99421 
99418 
y9415 
99412 
99409 
99406 
99402 
99399 
99396 
99393 
99390 
99386 
99383 
99380 
99377 
99374 
99370 
99367 
99364 
99360 
99357 
99354 
99351 
99347 
99344 
99341 
99337 
99334 
99331 
99327 
99324 
99320 
99317 
99314 
99310 
99307 
99303 
99300 
99297 
99293 
99290 
99286 
99283 
99279 
99276 
99272 
99269 
99205 
99262 
99268 
i>9255 
N.  COS.  N.sine, 


0463 

0482 

0511 

0540 

0569 

0.39'- 

0626 

0655 

0684 

0713 

0742 

0771 

0800 

0829 

0858 

0887 

0916 

0945 

0973 

002 

031 

060 

089 

118 

147 

176 

205 

234 

263 

291 

320 

349 

378 

407 

436 

465 

494 

523 

652 

580 

609 

638 

667 

696 

725 

754 

783 

812 

840 

869 

898 

927 

956 

985 

2014 

2043 

2071 

2100 

2129 

2158 

2187 


28 


Log.  Sines  and  Tangents.  (7°)  Natural  Sines. 


TABLE  n. 


D.  I0"j  Cosine. 


Sine. 


9.085894 
086922 
087947 
088970 
089990 
091008 
092024 
093037 
094047 
095056 
096062 
9.097065 
0980S6 
099065 
100062 
101056 
102048 
103037 
104025 
105010 
105992 

9.106973 
107951 
108927 
109901 
110873 
111842 
112809 
113774 
114737 
115698 

9.116656 
117613 
118567 
119519 
120469 
-121417 
122362 
123306 
124248 
126187 

9.126125 
127060 
127993 
128925 
129864 
130781 
131706 
132630 
133551 
134470 


51  9.135387 


136303 
137216 
138128 
139037 
139944 
140850 
141754 
142655 
143555 
Cosine. 


171 
171 
170 
170 
170 
169 
169 
168 
168 
168 
167 
167 
166 
166 
166 
165 
165 
164 
164 
164 
163 
163 
163 
162 
162 
162 
161 
161 
160 
160 


.996751 
996735 
996720 
996704 


996673 
996657 
996641 
996625 
996610 
996694 
9.996578 
996562 
996546 
996530 
996514 


996482 
996466 
996449 
996433 
9.996417 
996400 
996384 
996368 
996351 
996336 
996318 
996302 
996285 
,f.(.  996269 
i^n  9.996252 
996236 
996219 
996202 
996186 
996168 
996161 
996134 
996117 
996100 
996083 
996066 
996049 
996032 
996015 
995998 


169 
159 
159 
168 
168 
168 
157 
167 
157 
156 
156 
156 
155 
155 
164 
154 
154 
163 
163 
153 
152 
162 
152 
152 
161 
161 
151 
160 
150 


D.  lU' 


995963 
995946 
995928 
.995911 
995894 
996876 
995869 
995841 
995823 
995806 
995788 
996771 
995753 


Sine. 


2.6 

2.6 

2.6 

2.6 

2.6 

2.6 

2.6 

2.6 

2.6 

2.6 

2.6 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.8 

2.8 

2.8 

2.8 

2.8 

2.8 

2.8 

2.8 

2.8 

2.8 

2.8 

2.8 

2.8 

2.9 

2.9 

2.9 

2.9 

2.9 

2.9 

2.9 

2.9 

2.9 

2.9 

2.9 

2.9 

2.9 

2.9 

2.9 

2.9 

2.9 

2.9 

2.9 


Tang. 


9. 


.089144 

090187 

091228 

092286 

093302 

094336 

095367 

096395 

097422 

098446 

099468 

.100487 

101504 

102519 

103532 

104642 

105650 

106556 

107559 

108660 

109659 

.110556 

111561 

112543 

113533 

114521 

115507 

116491 

117472 

118452 

119429 

.120404 

121377 

122348 

123317 

124284 

126249 

126211 

127172 

128130 

129087 

130041 

130994 

131944 

132893 

133839 

134784 

136726 

136667 

137605 

138642 

139476 

140409 

141340 

142269 

143196 

144121 

145044 

145966 

146886 

147803 


Cotang. 


D.  JO^ 

174 

173 

173 

173 

172 

172 

171 

171 

171 

170 

170 

169 

169 

169 

168 

168 

168 

167 

167 

166 

166 

166 

165 

165 

165 

164 

164 

164 

163 

163 

162 

162 

162 

161 

161 

161 

160 

160 

160 

159 

159 

159 

168 

158 

158 

157 

157 

157 

156 

156 

156 

165 

165 

155 

154 

164 

164 

153 

153 

153 


Uotaug.   N.  sin(/ 


10.910856 
909813 
908772 
907734 
906698 
905664 
904633 
903605 
902578 
901554 
900532 

10.899513 
898496 
897481 
896468 
895458 
894450 
893444 
892441 
891440 
890441 

10.889444 
888449 
887467 
886467 
885479 
884493 
883509 
882528 
881648 
880571 

10.879596 
878623 
877652 
876683 
875716 
874751 
873789 
872828 
871870 
870913' 

10.869959; 
869006 
868056 
867107 
866161 
865216 
864274  I 
863333 
862395 
861458 
10.860524 
859591 
858660 
857731 
856804 
855879 ; 
864956 
854034  I 
853116 
852197 


;:  1218-/ 

i  12216 

l!  12245 

it  12274 

;  112302 

1 112331 

;  112360 

j 112389 

'1 12418 

ii  12447 

j 1 12476 

i|  12504 

1112533 

!l  12562 

1112591 

Ji  12620 

i|  12649 

112678 

12706 

12735 

12764 

12793 

12822 

12851 

12880 

1 12908 

12937 

12966 

1299 

13024 

13053 

13081 

13110 

13139 

13168 

1319 

13226 

! 13254 

; 13283 

13312 

! 13341 

13370 

1 13399 

i 13427 

j 1345G 

' 13485 

113514 

i 13543 

13572 

13600 

13629 

13658 

13687 

1371G 

13744 

13773 

; 13802 

i 13831 

1 13860 

! 13889 

113917 


N.  COK. 


99256 
99251 
99248 
99244 
99240 
99237 
9233 
99230 
99226 
99222 
99219 
99215 
99211 
99208 
99204 
99200 
99197 
99193 
99189 
99186 
99182 
99178 
99176 
99171 
99167 
99163 
99160 
99166 
99152 
99148 
99144 
99141 
99137 
99133 
99129 
99125 
99122 
99118 
99114 
99110 
99106 
)9102 
99098 
99094 
99091 
99087 
99083 
99079 
99075 
99071 
99067 
^9063 
:>y059 
139055 
99051 
:;9047 
J9043 
99039 
99035 
99031 
99027 


Tang.   liN.  C0.1.  .V.sine. 


82  Degrees. 


TABLE  II. 


Log.  Sines  and  Tangents.    (8°)    Natural  Sines. 


29 


Sine. 


9.143655 
144453 
145349 
146243 
147136 
148026 
148915 
149802 
150886 
151569 
152451 

9.153330 
154208 
156083 
155957 
156830 
167700 
158569 
159436 
160301 
161164 

9.162026 
162885 
163743 
164600 
165454 
166307 
167159 
168008 
168856 
169702 

9.170547 
171389 
172230 
173070 
173908 
174744 
175578 
176411 
177242 
178072 

9.178900 
179726 
180551 
181374 
182196 
183016 
183834 
184651 
186466 
186280 

9.187092 
187903 
188712 
189619 
190326 
191130 
191933 
192734 
193634 
194332 


D.  10" 


150 
149 
149 
149 
148 
148 
148 
147 
147 
147 
147 
146 
146 
146 
145 
146 
146 
144 
144 
144 
144 
143 
143 
143 
142 
142 
142 
142 
141 
141 
141 
140 
140 
140 
140 
139 
139 
139 
139 
138 
138 
138 
137 
137 
137 
137 
136 
136 
136 
136 
136 
136 
135 
135 
134 
134 
134 
134 
133 
133 


Cosine. 


Cosine. 

.995753 
996736 
996717 
995699 
995681 
995664 
996646 
996628 
995610 
995591 
996573 
.995556 
995537 
996619 
995601 
995482 
995464 
995446 
995427 
996409 
996390 

.996372 
996353 
995334 
996316 
996297 
995278 
995260 
995241 
995222 
995203 

.995184 
995165 
995146 
995127 
996108 
995089 
995070 
996061 
995032 
995013 

.994993 
994974 
994955 
994935 
994916 
994896 
994877 
994857 
994838 
994818 

.994798 
994779 
994759 
994739 
994719 
994700 
994680 
994660 
994640  I 
994620 : 
Sinel   I 


D.  lU" 


3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3,2 
3.2 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 


3.3 
3.3 
3.3 
3.3 


Tang. 


). 147803 
148718 
149632 
150644 
151454 
152363 
163269 
154174 
156077 
155978 
156877 

). 157776 
168671 
159566 
160467 
161347 
162236 
163123 
164008 
164892 
166774 

>.  166664 
167532 
168409 
169284 
170167 
171029 
171899 
172767 
173634 
174499 

M 76362 
176224 
177084 
177942 
178799 
179656 
180508 
181360 
182211 
183069 

1.183907 
184752 
186597 
186439 
187280 
188120 
188958 
189794 
190629 
191462 

1.192294 
193124 
193953 
194780 
195606 
196430 
197253 
198074 
198894 
199713 

Co  tang. 


D,  10' 


153 
152 
162 
152 
151 
151 
151 
150 
150 
160 
160 
149 
149 
149 
148 
148 
148 
148 
147 
147 
147 
146 
146 
146 
145 
146 
146 
145 
144 
144 
144 
144 
143 
143 
143 
142 
142 
142 
142 
141 
141 
141 
141 
140 
140 
140 
140 
139 
139 
139 
139 
138 
138 
138 
138 
137 
137 
137 
137 
136 


Cotang.  j  N.  sine 


10.852197 

851282 
850368 
849456 
848546 
847637 
846731 
846826 
844923 
844022 
843123 

10.842226 
841329 
840436 
839643 
838653 
837764 
836877 
835992 
835108 
834226 

10.833346 
832468 
831691 
830716 
829843 
828971 
828101 
827233 
826366 
825501 

10.824638 
823776 
822916 
822058 
821201 
820345 
819492 
818640 
817789 
816941 

10.816093 
815248 
814403 
813561 
812720 
811880 
811042 
810206 
809371 
808638 

10.807706 
806876 
806047 
805220 
804394 
803570 
802747 
801926 
801106 
800287 


3917 
3946 
3975 
4004 
4033 
4061 
4090 
4119 
4148 
4177 
4205 
42S4 
4263 
4292 
4320 
4349 
4378 
4407 
4436 
4464 
4493 
4522 
4561 
4580 
4608 
4637 
4666 
4695 
4723 
4752 
4781 
4810 
4838 
4867 
4896 
4926 
4954 
4982 
5011 
5040 
5069 
5097 
5126 
5155 
5184 
521^ 
5241 
5270 
5299 
532/ 
5356 
5585 
5414 
544'^ 
5471 
5500 
6629 
5557 
5586 
5616 
564Jj 


Tang. 


N.  cos.  N.sine 


60 
69 

68 
67 
66 
55 
64 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
36 
34 
33 
32 
31 
SO 
29 
28 
27 

98884  r26 

98880 

98876 

98871 

98867 

98863 

98858 

98854 

98849 

98845 

98841 

988a6 

98832 

9882-; 

98823 

98818 

98814 

98809 

98805 

98800 

98';  96 

98791 

98787 

98782 

98778 

98773 

98769 


99027 
99023 
99019 
99015 
99011 
99006 
99002 
98998 
98994 
98990 
98986 
98982 
98978 
98973 
98969 
98965 
98961 
98957 
98953 
98948 
98944 
98940 
98936 
98931 
98927 
98923 
98919 
98914 
98910 
98906 
98902 
98897 
98893 


81  Degrees. 


30 


Log.  Sines  and  Tangents.     (9°)     Natural  Sines. 


TABLE  n. 


Sine. 


9.194332 
195129 
195925 
196719 
197511 
198302 
199091 
199879 
200666 
201451 
202234 

9.203017 
203797 
204577 
205354 
206131 
206906 
207679 
208452 
209222 
209992 

9.210760 
211526 
212291 
213055 
213818 
214579 
215338 
216097 
216854 
217609 

9.218363 
219116 
219868 
220618 
221367 
222115 
222861 
223608 
224349 
225092 

9.225833 
226573 
227311 
228048 
228784 
229518 
230252 
230984 
231714 
232444 

^.233172 
233899 
234625 
235349 
236073 
236795 
237515 
238235 
238953 
239670 


Cosine. 


D.  10' 


133 
133 
132 
132 
132 
132 
131 
131 
131 
131 
130 
130 
130 
130 
129 
129 
129 
129 
128 
128 
128 
128 
127 
127 
127 
127 
127 
126 
126 
126 
126 
125 
125 
125 
125 
125 
124 
124 
124 
124 
123 
123 
123 
123 
123 
122 
122 
122 
122 
122 
121 
121 
121 
121 
120 
120 
120 
120 
120 
119 


Cosine. 


.994620 
994600 
994580 
994560 
994540 
994519 
994499 
994479 
994459 
994438 
994418 

.994397 
994377 
994367 
994336 
994316 
994295 
994274 
994254 
994233 
994212 

.994191 
994171 
994150 
994129 
994108 
994087 
994066 
994045 
994024 
994003 

.993981 
993960 
993939 
993918 
993896 
993875 
993854 
993832 
993811 
993789 

.993768 
993746 
993725 
993703 
993681 
993660 
993638 
993616 
993594 
993672 

.993550 
994528 
993506 
993484 
993462 
933440 
993418 
993396 
993374 
993361 


Sine. 


D.  10' 


3.3 
3.3 
3.3 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.5 
3.5 
3.5 
3.5 
3.5 
3.5 
3.5 
3.5 
3.5 
3.5 
3.5 
3.5 
3.6 
3.5 
3.6 
3.6 
3.5 
3.5 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 


3.7 
3.7 
3.7 
3.7 
3.7 
3.7 
3.7 
3.7 
3.7 
3.7 


Tans 


.199713 
200529 
201345 
202159 
202971 
203782 
204592 
205400 
206207 
207013 
207817 

.208619 
209420 
210220 
211018 
211815 
212611 
213405 
214198 
214989 
215780 

.216568 
217356 
218142 
218926 
219710 
220492 
221272 
222052 
222830 
223606 

.224382 
225156 
225929 
226700 
227471 
228239 
229007 
229773 
230539 
231302 

.232066 
232826 
233586 
234345 
235103 
235859 
236614 
237368 
238120 
238872 

.239622 
240371 
241118 
241865 
242610 
243354 
244097 
244839 
245579 
246319 


136 
136 
136 
135 
135 
135 
135 
134 
134 
134 
134 
133 
133 
133 
133 
133 
132 
132 
132 
132 
131 
131 
131 
131 
130 
130 
130 
130 
130 
129 
129 
129 
129 
129 
128 
128 
128 
128 
127 
127 
127 
127 
127 
126 
126 
126 
126 
126 
125 
125 
125 
125 
125 
124 
124 
124 
124 
124 
123 
123 


Cotang. 


Cotang.     I  IN.  sine.  N.  cos 


10.800287 
799471 
798655 
797841 
797029 
790218 
795408 
794600 
793793 
792987 
792183 

10.791381 
790580 
789780 
788982 
788185 
787389 
786595 
785802 
735011 
784220 

10.783432 
782644 
781858 
781074 
780290 
779508 
778728 
777948 
777170 
776394 

10.775618 
774844 
774071 
773300 
772529 
771761 
770993 
770227 
769461 
768698 

10.767935 
767174 
766414 
765655 
764897 
764141 
763386 
762632 
761880 
761128 

10.760378 
759629 
758882 
758135 
757390 
756646 
755903 
755161 
754421 
753681 


15643 
16672 
16701 
16730 
15758 
15787 
15816 
15845 
15873 
15902 
15931 
15959 
15988 
16017 
16046 
16074 
16103 
16132 
16160 
16189 
16218 
16246 
16275 
16304 
16333 
16361 
16390 
16419 
16447 
16476 
16505 
16533 
16562 
16691 
16620 


98769 
98764 
98760 
98755 
98751 
98746 
98741 
98737 
98732 
98728 
98723 
98718 
98714 
98709 
98704 
98700 
98695 
98690 
98686 
98681 
98676 
98671 
98667 
98662 
98657 
98652 
98648 
98643 
98638 


98629 
98624 
98619 
98614 
98609 


116648  98604 
!|  16677  98600 
J!  16706  98596 
116734  98590 
116763  .98585 
1116792  98580 
ii  16820  98575 
i  1 16849198570 
l'l6878|98565 
i  1690Gi9S561 
;  116935,98556 
16964  98561 
i;16992!98646 
17021  !98o41 
|17050'98536 
;17078!9S531 
:17107;9S526 
I  17136,98521 
Ii  17164198516 
i  17193198511 
i!  17222  98506 
ii  17250  98.501 


17279 
17308 
17336 
17365 


Tang. 


N.  cos.  N 


98496 
98491 
98486 
98481 


80  Degrees. 


TABLE  II. 


Log.  Sines  and  Tangents.    (10°)    Natural  Sines. 


31 


Sine. 


9.239670 
240386 
241101 
241814 
242626 
243237 
243947 
244666 
245363 
246069 
246775 

9.247478 
248181 
248883 
249583 
250282 
250980 
251677 
252373 
253067 
263761 
254453 
255144 
255834 
256523 
257211 
257898 
258583 
259268 
259951 
260633 
261314 
261994 
262673 
263351 
264027 
264703 
265377 
266051 
266723 
267395 

9.268065 
268734 
269402 
270069 
270735 
271400 
272064 
272726 
273388 
274049 
274708 
275367 
276024 
276681 
277337 
277991 
278644 
279297 
279948 
280599 


Cosine. 


D.  10" 


119 

119 

119 

119 

118 

118 

118 

118 

118 

117 

117 

117 

117 

117 

116 

116 

116 

116 

116 

116 

115 

115 

115 

115 

115 

114 

114 

114 

114 

114 

113 

113 

113 

113 

113 

113 

112 

112 

112 

112 

112 

11 

11 

11 

11 

11 

11 

110 

110 

110 

110 

110 

110 

109 

109 

109 

109 

109 

109 

108 


Cosine. 


,993351 
993329 
993307 
993285 
993262 
993240 
993217 
993195 
993172 
993149 
993127 

.993104 
993081 
993059 
993036 
993013 
992990 
992967 
992944 
992921 
992898 

.992875 
992852 
992829 
992806 
992783 
992769 
992736 
992713 
992690 
992666 

.992643 
992619 
992596 
992672 
992649 
992525 
992501 
992478 
992464 
992430 

.992406 
992382 
992369 
992335 
992311 
992287 
992263 
992239 
992214 
992190 

.992166 
992142 
992117 
992093 
992069 
992044 
992020 
991996 
991971 
991947 


Sine. 


D.  10" 


3.7 
3.7 
3.7 
3.7 
3.7 
3.7 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
4.0 
4.0 
4.0 
4.0 
4.0 
4.0 
4.0 
4.0 
4.0 
4.0 
4.0 
4.0 
4.0 
4.0 
4.0 
4.1 
4.1 
4.1 
4.1 
4.1 
4.1 
4.1 


Tang. 


.246319 
347057 
247794 
248530 
249264 
249998 
250730 
251461 
252191 
252920 
253648 

.254374 
255100 
255824 
256547 
267269 
267990 
258710 
259429 
260146 
260863 

.261678 
262292 
263005 
263717 
264428 
266138 
265847 
266555 
267261 
267967 

1.268671 
269376 
270077 
270779 
271479 
272178 
272876 
273573 
274269 
274964 

'.275658 
276351 
277043 
277734 
278424 
279113 
279801 
280488 
281174 
281868 

1.282542 
283225 
283907 
284588 
285268 
285947 
286624 
287301 
287977 
288662 

Cotang. 


D.  10" 


123 
123 
123 
122 
122 
122 
122 
122 
121 
121 
121 
121 
121 
120 
120 
120 
120 
120 
120 
119 
119 
119 
119 
119 
118 
118 
118 
118 
118 
118 
117 
117 
117 
117 
117 
116 
116 
116 
116 
116 
116 
115 
115 
115 
115 
115 
115 
114 
114 
114 
114 
114 
114 
113 
113 
113 
113 
113 
113 
112 


Cotang. 


10.763681 
762943 
752206 
761470 
750736 
750002 
749270 
748539 
747809 
747080 
746352 

10.745626 
744900 
744176 
743463 
742731 
742010 
741290 
740571 
739854 
739137 

10.738422 
737708 
736995 
736283 
736572 
734862 
734163 
733445 
732739 
732033 

10.731329 
730626 
729923 
729221 
728521 
727822 
727124 
726427 
726731 
726036 

10.724342 
723649 
722967 
722266 
721576 
720887 
720199 
719512 
718826 
718142 

10.717468 
716776 
716093 
715412 
714732 
714053 
713376 
712699 
712023 
711348 


N.sine.  N.  cos 


Tang. 


17365 
17393 
17422 
17451 
17479 
17508 
17537 
17665 
17694 
17623 
17661 
17680 
17708 
17737 
17766 
17794 
17823 
17852 


98481 
98476 
98471 
98466 
98461 
98455 
98450 
98445 
98440 
98435 
98430 
98425 
98420 
98414 
98409 
98404 
98399 
98394 


17880  98389 


17909 
17937 
17966 
17995 
18023 
18052 
18081 
18109 
18138 
18166 
18195 
18224 
18252 
18281 
18309 
18338 
18367 
18395 
18424 
18452 
18481 
18509 


98383 
98378 
98373 
98368 
98362 
98357 
98352 
98347 
98341 
98336 
98331 
98325 
98320 
98315 
98310 
98304 
98299 
98294 
9828S 
98283 
98277 
98272 


18538198267 
1866798261 
1869598256 
18624  98250 


18652 
18681 
18710 
18738 
18767 


98245 
98240 
98234 
98229 
98223 


18796  98218 
18824  98212 
1885298207 
1888198201 
1891098196 
18938  98190 
18967I9S185 
1899598179 
19024198174 
19052198168 
1908198163 


N.  COS.  N.Fine. 


79  Degrees. 


32 


Log.  Sines  and  Tangents.    (11°)    Natural  Sines. 


TABLE  II. 


Sine. 


DTTi)^ 


280599 
281248 
281897 
282544 
283190 
283836 
284480 
285124 
285766 
286408 
287048 
287687 
288326 
288964 
289600 
290236 
290870 
291504 
292137 
292768 
293399 

9.294029 
294658 
295286 
295913 
296539 
297164 
297788 
298412 
299034 
299655 

9.300276 
300895 
301514 
302132 
302748 
303364 
303979 
304593 
305207 
305819 

9.306430 
307041 
307650 
308259 
308867 
309474 
310080 
310685 
311289 
311893 
312495 
313097 
313698 
314297 
314897 
315495 
316092 
316689 
317284 
317879 


Cosine. 


108 
108 
108 
108 
108 
107 
107 
107 
107 
107 
107 
106 
106 
106 
106 
106 
106 
105 
105 
105 
105 
105 
105 
104 
104 
104 
104 
104 
104 
104 
103 
103 
103 
103 
103 
103 
102 
102 
102 
102 
102 
102 
102 
101 
101 
101 
101 
101 
101 
100 
100 
100 
100 
100 
100 
100 
100 
99 
99 
99 


Cosine.  ID.  lu 


.991947 
991922 
991897 
991873 
991848 
991823 
991799 
991774 
991749 
991724 
991699 

L 991674 
991649 
991624 
991599 
991574 
991549 
991524 
991498 
991473 
991448 

'.99L422 
991397 
991372 
991346 
991321 
991295 
991270 
991244 
991218 
991193 

.991167 
991141 
991115 
991090 
991064 
991038 
991012 
990986 
990960 
990934 
990908 
990882 
990855 
990829 
990803 
990777 
990750 
990724 
990697 
990671 

1.990644 
990618 
990591 
990566 
990538 
990511 
990485 
990458 
990431 
990404 


Sine. 


4.1 
4.1 
4.1 
4.1 
4.1 
4.1 
4.1 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.5 
4.5 
4.5 
4.5 


Tiiug.   ID.  iv 


1.288652 
289326 
289999 
290671 
291342 
292013 
292682 
293360 
294017 
294684 
295349 

►  .296013 
296677 
297339 
298001 
298662 
299322 
299980 
300638 
301295 
301951 

1.302607 
303261 
303914 
304667 
305218 
305869 
306519 
307168 
307815 
308463 

•  .309109 
309754 
310398 
311042 
311685 
312327 
312967 
313608 
314247 
314885 

1.316623 
316159 
316795 
317430 
318064 
318697 
319329 
319961 
320592 
321222 

1.321851 
322479 
323106 
323733 
324358 
324983 
325607 
326231 
326853 
327476 

Cotang. 


112 
112 
112 
112 
112 
111 
111 
111 
111 
111 
111 
111 
110 
110 
110 
110 
110 
110 
109 
109 
109 
109 
109 
109 
109 
108 
108 
108 
108 
108 
108 
107 
107 
107 
107 
107 
107 
107 
106 
106 
106 
106 
106 
106 
106 
105 
105 
105 
106 
105 
105 
105 
104 
104 
104 
104 
104 
104 
104 
104 


lug.  jN.  sine.  N.  cos 


10.711348 
710674 
710001 
709329 
708658 
707987 
707318 
706650 
705983 
705316 
704651 

10.703987 
703323 
702661 
701999 
701338 
700678 
700020 
699362 
698705 
698049 

10.697393 
696739 
696086 
696433 
694782 
694131 
693481 
692832 
692185 
691637 

10-690891 
690246 
689602 
688958 
688315 
687673 
687033 
686392 
686753 
686115 

10-684477 
683841 
683205 
682670 
681936 
681303 
680671 
680039 
679408 
678778 

10.678149 
677521 
676894 
676267 
675642 
675017 
674393 
673769 
673147 
672526 


11908198163 
!  19109  98157 


119138 

i 19167 
19195 
19224 

1 19252 
19281 

' 19309 
19338 
19366 


19395  98101 


19423 
19452 
19481 
19509 
19538 
19566 
19595 
19623 
19652 
19680 
19709 
19737 
19766 
197^4 
19823 
19861 
19880 
1 19908 
19937 
19965 
! 19994 
1 20022 
•20061 
'20079 
20108 
20136 
20166 
20193 


20260 
20279 
20307 
20336 
20364 
20393 
20421 
{20450 
120478 
1 20507 
120535 
1 20563 
20592 
1 20620 
120649 
i 20677 
1 20706 
120734 
i 2076a 
120791 


Tang. 


98152 
98146 
98140 
98135 
98129 
98124 
98118 
98112 
98107 


98096 
98090 
98084 
98079 
98073 
98067 
98061 
98056 
98050 
98044 
98039 
98033 
98027 
98021 
98016 
98010 
98004 
97998 
97992 
97987 
97981 
97975 
97969 
97963 
97958 
97952 
97946 
97940 


20222(97934 


97928 
97922 
97916 
97910 
97905 
97899 
97893 
97887 
97881 
97875 
97869 
9/863 
97857 
97851 
97846 
97839 
97833 
97827 
97821 
97815 
N.  cos.  N.pine 


78  Degrees. 


TABLE  II. 


Log.  Sines  and  Tangents.    (12°)    Natural  Sines. 


33 


0 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
;  44 
!  45 
I  46 
ii  47 
48 
i  49 
50 
I  51 
62 
53 
54 
55 
56 
57 
58 
59 
60 


Sine. 

9.317879 
318473 
3190i)ti 
319658 
320249 
320840 
321430 
322019 
322607 
323194 
323780 

9.324366 
324950 
325534 
326117 
326700 
327281 
327862 
328442 
329021 
329599 

9.330176 
330753 
331329 
331903 
332478 
333051 
333624 
334195 
334766 
335337 

9.335906 
336475 
337043 
337610 
338176 
338742 
339306 
339871 
340434 
340996 

9.341558 
342119 
342679 
343239 
343797 
344355 
344912 
345469 
346024 
346579 
19.347134 
347687 
348240 
348792 
349343 
349893 
350443 
330992 
351540 
352088 


D.  10" 


Cosine. 


99.0 

98.8 

98.7 

98.6 

98.4 

98.3 

98.2 

98.0 

97.9 

97.7 

97.6 

97.5 

97.3 

97.2 

97.0 

96.9 

96.8 

96.6 

96.5 

96.4 

96.2 

96.1 

96.0 

95.8 

95.7 

95.6 

95.4 

95.3 

95.2 

95.0 

94.9 

94.8 

94.6 

94.5 

94.4 

94.3 

94.1 

94.0 

93.9 

93.7 

93.6 

93.5 

93.4 

93.2 

93.  i 

93.0 

92.9 

92.7 

92.6 

92.5 

92.4 

92.2 

92.1 

92.0 

91.9 

91.7 

91.6 

91.5 

91.4 

91.3 


Cosine. 

>.  990404 
990378 
990351 
990324 
990297 
990270 
990243 
990215 
990188 
990161 
990134 

1.990107 
990079 
990052 
990025 
989997 
989970 
989942 
989915 
989887 
989860 

1.989832 
989804 
989777 
989749 
989721 
989693 
989665 
989637 
989609 
989582 
.989553 
989525 
989497 
989469 
989441 
989413 
989384 
989356 
989328 
989300 

.989271 
989243 
989214 
989186 
989157 
989128 
989100 
989071 
989042 
989014 
9.988985 
988956 
988927 
988898 
988869 
988840 
988811 
988782 
988753 
988724 


Sine. 


D.  10" 


4.5 
4.5 
4.5 


4 

4 

4 

4 

4, 

4 

4 

4, 

4, 

4.6 

4.6 

4.6 

4.6 

4.6 

4.6 

4.6 

4.6 

4.6 

4.6 

4.6 

4.6 

4,7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.7 

4.8 

4.8 

4.8 

4.8 

4.8 

4.8 

4.8 

4.8 

4.8 

4.8 

4.8 

4.9 

4.9 

4.9 


Tang. 

9.327474 
3280^5 
328715 
329334 
329953 
330570 
33118 
331803 
332418 
333033 
333646 

9.334259 
334871 
335482 
336093 
336702 
337311 
337919 
338527 
339133 
339739 

9.340344 
340948 
341552 
342155 
342757 
343358 
343968 
344558 
345157 
345755 

9.346353 
346949 
347546 
348141 
348735 
349329 
349922 
350514 
351106 
351697 

9.352287 
352876 
353465 
354053 
354640 
355227 
355813 
356398 
356982 
357566 

9.358149 
358731 
359313 
359893 
360474 
361053 
301632 
362210 
362787 
363364 
Cotang. 


D.  lU" 


103 

103 

103 

103 

103 

103 

103 

102 

102 

102 

102 

102 

102 

102 

102 

101 

101 

101 

101 

101 

101 

101 

101 

100 

100 

100 

100 

100 

100 

100 

100 
99.4 
99.3 
99.2 
99.1 
99.0 
98. 
98.7 
98.6 
98.5 
98.3 
98.2 
98.1 
98.0 
97.9 
97.7 
97.6 
97.5 
97.4 
97.3 
97.1 
97.0 
96.9 
90.8 
96.7 
96.6 
96.6 
96.3 
96.2 
96.1 


Gotaag. 


10 


10 


10 


10 


10 


10 


.672526 
671905 
671285 
670866 
670047 
669430 
668813 
668197 
667582 
666967 
666354 
.665741 
665129 
664518 
663907 
663298 
662689 
662081 
661473 
660867 
660261 
.659666 
659052 
658448 
657845 
657243 
656642 
656042 
655442 
654843 
654245 
.653647 
653051 
652456 
651859 
651263 
650671 
650078 
649486 
648894 
648303 
.647713 
647124 
646535 
646947 
645360 
644773 
644187 
648602 
643018 
642434 
.641851 
641269 
640687 
640107 
639626 
638947 
638368 
637790 
637213 
636636 


N.  8!Ue.  x\.  COK. 


20791  97815 

20820  97809 

20848  97803 

20877  97797 

20905  97791 

20933  97784 

20962  97778 

20990  97772 

21019  97766 

21047  97760 

21076  97764 

21104  97745 

21132  97742 

21161  97735 

21189  97729 

21218  97723 

!  21246  97717 

21275  97711 

21303  97705 

2133197698 

21360  97692 

21388  97686 

21417:97680 

21445  97673 

2147497667 

21502  97661 

21530  97655 

21569  97648 

21587  97642 

2161697636 

2164497630 

21672:97623 

21701 '97617 

21729  97611 

21758 '97604 

|i2178a!97598 

!  21814:97592 

121843  97585 

;i2187l!97579 

1121899:97573 

;|21928'97566 

!  1 21956:97660 

!|21985;97653 

!:22013!97547 

1^2204197541 

1 22070,97534 

!i2209b!97528 

|i  22126:97521 

j;  22155  97515 

i  22183  97608 

!' 22212 '97502 

i  22240197496 

[:  22268 197489 

i' 2229  7 197483 

1 122326 197476 

'122353  97470 

!  122382  97463 


122410 

i  22438 
! 22467 
I  22495 


Tang. 


N.  cos.  N.sine 


97457 
97450 
97444 
97437 


77  Degrees. 


34 


Log.  Sines  and  Tangents.    (13°)    Natural  Sines. 


TABLE  IL 


Sine. 

352088 
352635 
353181 
353726 
354271 
354815 
355358 
355901 
356443 
356984 
357524 
358064 
358603 
359141 
359678 
360215 
360752 
361287 
361822 
362356 
362889 

9.363422 
363954 
364485 
365016 
365546 
366075 
366604 
367131 
367669 
368185 

9.368711 
369236 
369761 
370286 
370808 
371330 
371852 
372373 
372894 
373414 

9.373933 
374452 
374970 
376487 
376003 
376519 
377036 
377549 
378063 
378677 

9.379089 
379601 
380113 
380624 
381134 
381643 
382162 
382661 
383168 
383676 


C)osine. 


D.  10''     Cosine.     D.  10"      Tang.      D.  10"     Cotang.     |  N.sine  IN.  cos, 


91.1 
91.0 
90.9 
90.8 
90.7 
90.5 
90.4 
90.3 
90.2 
90.1 
89.9 
89.8 
89.7 
89.6 
89.5 
89.3 
89.2 
89.1 
89.0 
88.9 
88.8 
88.7 
88.5 
88.4 
88.3 
88.2 
88.1 
88.0 
87.9 
87.7 
87.6 
87.6 
87.4 
87.3 
87.2 
87.1 
87.0 
86.9 
86.7 
86.6 
86.6 
86.4 
86.3 
86.2 
86.1 
86.0 
86.9 
85.8 
86.7 
85.6 
85.4 
85.3 
85.2 
85.1 
86.0 
84.9 
84.8 
84.7 
84.6 
84.5 


.988724 
988695 
988666 


988607 
988578 
988548 
988519 
988489 
988460 
988430 
.988401 
988371 
988342 
988312 
988282 
988352 
988223 
988193 
988163 
988133 
.988103 
988073 
988043 
988013 
987983 
987953 
987922 
987892 
987862 
987832 
.987801 
987771 
987740 
987710 
987679 
987649 
987618 
987588 
987557 
987526 
.987496 
987466 
987434 
987403 
987372 
987341 
987310 
987279 
987248 
987217 
.987186 
987155 
987124 
987092 
987061 
987030 


986967 
986936 
986904 


Sme. 


4.9 
4.9 
4.9 
4.9 
4.9 
4.9 


9 
9 
9 
9 
9 
9 
4.9 
4.9 
5.0 
5.0 
5.0 
5.0 
5.0 
5.0 
6.0 
5.0 
6.0 
5.0 
6.0 
6.0 
5.0 
5.0 
6.0 
6.0 
5.1 
5.1 
5.1 
5.1 
5.1 
5.1 
5.1 
5.1 
5.1 
5.1 
5.1 
5.1 
6.1 
5.1 
5.2 
5.2 
6.2 
5.2 
6.2 
5.2 
5.2 
5.2 
5.2 
5.2 
5.2 
5.2 
6.2 
5.2 
5.2 
6.2 


9.363364 
363940 
364515 
365090 
365664 
366237 
366810 
367382 
367953 
368524 
369094 

9.369663 
370232 
370799 
371367 
371933 
372499 
373064 
373629 
374193 
374756 

9.376319 
375881 
376442 
377003 
377663 
378122 
378681 
379239 
379797 
380364 

9.380910 
381466 
382020 
382676 
383129 


384234 
384786 
385337 


9.386438 
386987 
387636 


388631 
389178 
389724 
390270 
390815 
391360 
9.391903 
392447 


393531 
394073 
394614 
395154 
395694 
396233 
396771 


Cotang. 
Degrees. 


10.636636 
636060 
635485 
634910 
634336 
633763 
633190 
632618 
632047  ; 
631476  j 
630906' 

10.630337 
629768  I 
629201 i 
628633 
628067' 
627501 
626936 
626371 
625807 
625244 

10.624681 
6241 19 
623558 
622997 
622437 
621878 
621319 
620761 
620203 
619646 

10.619090 
618534 
617980 
617425 
616871 
616318 
615766 
615214 
614663 
614112 

10.613562 
613013 
612464 
611916 
611369 
610822 
610276 
609730 
609185 
60SG40 

10.608097 
607553 
607011 
606469 
605927 
605386 
604846 
604306 
603767 
603229 


22495  97437 
22523  97430 
22552  97424 
2258097417 
22608  97411 
22637  97404 
22665  97398 
22693  97391 
22722  97384 
22750197378 


22778 
22807 
22836 


97371 
97365 
97358 


22863(97351 
22892197345 


Tang. 


22920 
22948 
22977 
23005 
23033 
23062 
23090 
23118 
28146 
23175 
23203 
23231 
23260 
23288 
23316 
23345 
23373 
23401 
23429 
23458 
23486 
23514 
23542 
23571 


23627 
23656 
23684 
23712 
23740 
23769 


23797  97127 
23826  97120 


23853 
23882 
23910 
23938 
23966 
23995 
24023 
24051 
24079 
24108 
24136 
24164 
2419-2 


97338 
97331 
97325 
97318 
97311 
97304 
97298 
97291 
97ii84 
97278 
97271 
97264 
97257 
97251 
97244 
97237 
97230 
97223 
97217 
97210 
97203 
97' 96 
97189 
97182 


23599  97176 


97169 
97162 
97156 
97148 
97141 
97134 


97113 
97106 
97100 
97093 

)iom 

97079 
97072 
97065 
97068 
97051 
97044 
7037 
97030 
>".  COS.  N.sine, 


TABLE  II. 


Log.  Sines  and  Tangents.    (14°)    Natural  Sines. 


35 


0 
1 

2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
65 
56 
57 
58 
69 
60 


Sine. 


1.383675 
384182 
384687 
385192 
385697 
386201 
386704 
387207 
387709 
388210 
388711 
.389211 
389711 
390210 
390708 
391206 
391703 
392199 
392695 
393191 
393685 
.394179 
394673 
395166 
395658 
396150 
396641 
397132 
397621 
398111 
398600 
.399088 
399575 
400062 
400549 
401035 
401520 
402005 
402489 
402972 
403455 
.403938 
404420 
404901 
406382 
405862 
406341 
406820 
407299 
407777 
408254 
.408731 
409207 
409682 
410157 
410632 
411106 
411579 
412052 
412524 
412996 


D.  10" 


Cosine. 


84.4 

84.3 

84.2 

84.1 

8i.0 

83.9 

83.8 

83.7 

83.6 

83.5 

83.4 

83.3 

83.2 

83.1 

83.0 

82.8 

82.7 

82.6 

82.5 

82.4 

82.3 

82.2 

82.1 

82.0 

81.9 

81.8 

81.7 

81.7 

81.6 

81.5 

81.4 

81.3 

81.2 

81.1 

81.0 

80.9 

80.8 

80.7 

80.6 

80.5 

80.4 

80.3 

80,2 

80.1 

80.0 

79.9 

79.8 

79.7 

7y.6 

79.5 

79.4 

79.4 

79.3 

79.2 

79.1 

79. U 

78.9 

78. b 

78.7 

78.6 


Cosine. 


1.986904 
986873 
986841 
986809 
986778 
986746 
986714 
986683 
986651 
986619 
986587 

1.986555 
986523 
986491 
986459 
986427 
986395 
986363 
986331 
986299 
986266 

.986234 
986202 
986169 
986137 
986104 
986072 
986039 
986007 
985974 
985942 

.985909 
985876 
985843 
985811 
985778 
985745 
985712 
985679 
985646 
986613 

.985580 
985547 
985514 
985-180 
985447 
985414 
985380 
985347 
985314 
985280 

.985247 
985213 
985180 
985146 
985113 
985079 
985045 
985011 
984978 
984944 


D.  10" 


Sine. 


5.2 
5.3 
6.3 
5.3 
6.3 
6.3 
5.3 
5.3 
5.3 
6.3 
5.3 
5.3 
6.3 
5.3 
6.3 
6.3 
5.3 
5.4 
6.4 
6.4 
5.4 
5.4 
6.4 
6.4 
5.4 
5.4 
5.4 
5.4 
5.4 
5.4 
5.4 
5.5 
5.5 
6.5 
5.5 
5.5 
5.5 
5.6 
5.6 
5.5 
5.5 
6.6 
6.5 
5.5 
6.5 
6.6 
6.6 
6.6 
6.6 
5.6 
5.6 
5.6 
5.6 
6.6 
6.6 
6.6 
5.6 
6,6 
5.6 
5.6 


Tang. 


9.396771 
397309 
397846 
398383 
398919 
399456 
399990 
400624 
401058 
401691 
402124 

9.402656 
403187 
403718 
404249 
404778 
405308 
405836 
406364 
406892 
407419 
407945 
408471 
408997 
409521 
410045 
410669 
411092 
411616 
412137 
412668 

9.413179 
413699 
414219 
414738 
416267 
416775 
416293 
416810 
417326 
417842 
418358 
418873 
419387 
419901 
420415 
420927 
421440 
421952 
422463 
422974 

9.423484 
423993 
424503 
425011 
425519 
426027 
426634 
427041 
427547 
428052 


D.  10' 


Co  tang. 


Cotans 


!N.  sine.  N.  cos 


10.603229 
602691 
602154 
601617 
601081 
600545 
600010 
599476 
598942 
698409 
697876 
10.697344 
696813 
696282 
595751 
596222 
694692 
694164 
593636 
693108 
692581 
10.592055 
591529 
591003 
690479 
689956 
589431 
588908 
588385 
687863 
587342 
10.686821 
586301 
585781 
585262 
584743  j 
684225 1 
583707  I 
583190  j 
58267411 
582168  I j 
10.5816421 1 
581127  || 
580613  I i 
68009911 
579685  I 
679073  I 
6785601 
578048  i 
677637  i 
677026  I 
10.676516 
576007 
676497 
574989 
674481 
573973!  I 
573466  I 
672969;! 
572463  1 1 
571948!  I 


24192 

24220 

24249 

24277 

24305 

24333 

24362 

24390 

24418 

24446 

24474 

24503 

24531 

24559 

24587 

24615 

24644 

24672 

24700 

24728 

24766 

24784 

24813 

24841 

24869 

24897 

24925 

24954 

24982 

25010 

25038 

26066 

26094 

26122 

26151 

25179 

2520 

25236 

25263 

25291 

25320 

25348 

25376 

26404 

25432 

25460 

25488 

26516 

25545 

25573 

25601 

25629 

26667 

26685 

25713 

26741 

25766 

25798 

25826 

26864 

26882 


97030 

97023 

97015 

97008 

97001 

96994 

96987 

96980 

96973 

96966 

96969 

96952 

96945 

96937 

96930 

96923 

96916 

96909 

96902 

96894 

96887 

96880 

96873 

96866 

96858 

96851 

96844 

96837 

96829 

96822 , 

96816 

96807 

96800 

96793 

96786 

96778 

96771 

96764 

96756 

96749 

96742 

96734 

96727 

96719 

96712 

96705 

96697 

96690 

96682 

96675 

96067 

96660 

96663 

96645 

96638 

96630 

96623 

96616 

96608 

96600 

96593 


Tang. 


N.  COS.  N.pine.  ' 


75  Degrees. 


20 


36 


Log.  Sines  and  Tangents.    (15°)    Natural  Sines. 


TABLE  II. 


Sine. 


412996 
413467 
413938 
414408 
414878 
415347 
416815 
416283 
416751 
417217 
417684 
418150 
418615 
419079 
419544 
420007 
420470 
420933 
421395 
421857 
422318 

9.422778 
423238 
4236y7 
424156 
424615 
425073 
425530 
425987 
426443 
426899 

9.427354 
427809 
428263 
428717 
429170 
429623 
430075 
430527 
430978 
431429 

9.431879 
432329 
432778 
433226 
433676 
434122 
434569 
435016 
435462 
435908 

9.436353 
436798 
437242 
437686 
438129 
438572 
439014 
439456 
439897 
440338 


Cosine. 


D.  10' 


70 

78 

78.0 

77.9 

77.8 

77.7 

77.6 

77.5 

77.4 

77.3 

77.3 

77.2 

77.1 

77.0 

76.9 

76.8 

76.7 

76.7 

76.6 

76.5 

76.4 

76.3 

76.2 

76.1 

76.0 

76.0 

75.9 

75.8 

75.7 

75.6 

75.5 

75 

75 

75 

75 

75 

75 

74.9 

74.9 

74.8 

74.7 

74  6 

74.5 

74.4' 

74.4 

74.3 

74.2 

74.1 

74.0 

74.0 

73.9 

73.8 

73.7 

73.6 

73.6 

73.6 


Cosine. 


.984944 
984910 
984876 
984842 
984808 
984774 
984740 
984706 
984672 
984637 
984603 
.984569 
984535 
984500 
984466 
984432 
984397 
984363 
984328 
984294 
984259 
.984224 
984190 
984156 
984120 
984085 
984050 
984015 
983981 
983946 
983911 
.983875 
983840 
983805 
983770 
983736 
983700 
983664 
983629 
983594 
983558 
.983523 
983487 
983452 
983416 
983381 
983345 
983309 
983273 
983238 
9S3202 
.983166 
983130 
983094 
983058 
983022 
982986 
982960 
982914 
982878 
982842 


Sine. 


D.  Wi      Tang. 


5.7 
5.7 
5.7 
5.7 
5.7 
5.7 
5.7 
5.7 
5.7 
6.7 
5.7 
6.7 
5.7 
5.7 
5.7 
6.8 
5.8 
6.8 
5.8 
5.8 
5.8 
5.8 
5.8 
5.8 
5.8 
5.8 
6.8 
5.8 
5.8 
5.8 
5.8 
5.8 
6.9 
6.9 
6.9 
5.9 
5.9 
6.9 
5.9 
5.9 
6.9 
5.9 
6.9 
6.9 
5.9 
6.9 
6.9 
5.9 
6.0 
6.0 
6.0 
6.0 
6.0 
6.0 
6.0 
6.0 
6.0 
6.0 
6.0 
6.0 


9.428052 
428557 
429062 
42956U 
430070 
430573 
431075 
431577 
432079 
432680 
433080 

9.433580 
434080 
434579 
435078 
435576 
436073 
436570 
437067 
437563 
438059 

9.438554 
439048 
439543 
440036 
440529 
441022 
441514 
442006 
442497 
442988 

9.443479 
443968 
444458 
444947 
445435 
445923 
446411 
446898 
447384 
447870 

9.448356 
448841 
449326 
449810 
450294 
460777 
451260 
461743 
452225 
462706 

9.463187 
453668 
464148 
454628 
455107 
455586 
456064 
450542 
457019 
457496 


Cotang. 


D.  10" 


Cotaug.   N.  sine, 

10.571948 

571443 

570938 

57U434 

569930 

51)94-27 

668925 

568423 

567921 

567420 

666920 
10.566420 


2688 

2591 U 

2593 

2596 

25994 

2602-i 

26050 

26079 

26107 

26135 

26163 

26191 


5659-20  26219 
665421  i  26247 
564922: 1 26275 
564424  I!  26303 
5C3927;;  26331 
6G3430  I  26359  96463 
562933!  26387 
662437!  126415 
561941  1126443 


26471 
26500 


10.561446 

660952 
560457 

659964  1 1  26556 
5594711!  26584 
558978  I 
558486!  126640 
557994!  {26668 
567503;  26696 
5570121:26724 

10.656521  !!  26752 
556032  jj 267 SO 
555542  : 1  2680b 
555053!  26836 
554566  1 126864 
654077!  I  26892 
553689!  126920 
653102  i{  2694b 
552616;  1 2697b 
552130:127004 

10.551644;  27032 
651159;  2;  000 
550674::  27088 
550190!' 27110 
549706  ii27l4-i 


549223 
548740 
548257 
547775 
647294 
10,546813 
546332 
545852 
545372 
544893 
644414 
543936 
543458 
642981 
642504 


27172 
27200 
2722b 
2725(> 
127284 
127312 
27340 
! 2736b 
127396 
I  27424 
j  27462 
127480 
27508 
27  — 
I  27564 


96593 
96585 
96578 
96570 
96562 
96565 
96547 
96540 
96632 
96624 
96617 
96509 
96502 
96494 
96486 
96479 
96471 


96456 
96448 
96440 
96433 
96425 


26528  96417 
96410 
96402 

26612  96394 
96386 
96379 
96371 
96363 
06355 
96347 
96340 
;36332 
96324 
96316 
96308 
96301 
96293 
96285 
96277 
96269 
962b  1 
96253 
96246 
96238 
96230 
96222 
96214 
96206 
96198 
96190 
96182 
96174 
96166 
96168 
96160 
96142 


636  96134 
96126 


Tang. 


N.  cos.lN.sine, 


74  Degrees. 


or  TH 


I.A 


£AUhO^^>^ 


TABLE  II. 


Log.  Sines  and  Tangents.    (16°)    Natural  Sines. 


37 


Sine.      D.  10"     Cosine.     D.  10"      Tang,      D.  10"     Cotang.    j  N.  sine.  N.  cos. 


.440338 
440778 
441218 
441658 
442096 
442535 
442973 
443410 
443847 
444284 
444720 

,445155 
445590 
446025 
446459 
446893 
447326 
447759 
448191 
448623 
449054 

.449485 
449915 
450345 
450775 
451204 
451632 
462060 
452488 
452915 
453342 

,453768 
454194 
454619 
455044 
455469 
455893 
456316 
456739 
457162 
457584 

.458006 
458427 
458848 
459208 
459688 
460108 
460527 
460946 
461364 
461782 

.462199 
462616 
463032 
463448 
463864 
464279 
464694 
466108 
465522 
465935 


73.4 
73.3 
73.2 
73-1 
73.1 
73.0 
72.9 
72.8 
72.7 
72.7 
72.6 
72.6 
72.4 
72.3 
72.3 
72.2 
72.1 
72.0 
72.0 
71.9 
71.8 
71.7 
71.6 
71.6 
71.5 
71.4 
71.3 
71.3 
71.2 
71.1 
71.0 
71.0 
70.9 
70.8 
70.7 
70.7 
70.6 
70.5 
70.4 
70.4 
70.3 
70.2 
70.1 
70.1 
70.0 
69.9 
69.8 
69.8 
69.7 
69.6 
69.5 
69.5 
69.4 
69.3 
69.3 
69.2 
69.1 
69.0 
69.0 
68.9 


.982842 
982805 
982769 
982733 
982696 
982660 
982624 
982587 
982551 
982614 
982477 

.982441 
982404 
982367 
982331 
982294 
982257 
982220 
982183 
982146 
982109 

.982072 
982035 
981998 
981961 
981924 
981886 
981849 
981812 
981774 
981737 

.981699 
981662 
081625 
981587 
981549 
981512 
981474 
981436 
981399 
981361 

.981323 
981285 
981247 
981209 
981171 
981133 
981095 
981057 
981019 
980981 

',980942 
980904 
980866 
980827 
980789 
980750 
980712 
980673 
980636 
980596 


Sine. 


6.0 
6.0 
6.1 
6.1 
6.1 
6.1 
6.1 
6.1 
6.1 
6.1 
6.1 
6.1 
6.1 
6.1 
6.1 
6.1 
6.1 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.4 
6.4 
6.4 
6.4 
6.4 
6.4 
6.4 
6.4 
6.4 
6.4 
6.4 
6.4 
6.4 
6.4 


9.457496 
457973 
458449 
468925 
459400 
469875 
460349 
460823 
461297 
461770 
462242 

9.462714 
463186 
463658 
464129 
464699 
466069 
466639 
466008 
466476 
466946 

9  467413 
'467880 
468347 
468814 
469280 
469746 
470211 
470676 
471141 
471606 

9  472068 
'472532 
472995 
473457 
473919 
474381 
474842 
476303 
475763 
476223 

9  476683 
477142 
477601 
478059 
478517 
478975 
479432 
479889 
480345 
■  480801 

9  481257 
481712 
482167 
482621 
483076 
483529 
483982 
484436 
484887 
485339 


Cotang. 


10 


10 


10 


10 


10 


10 


542604 
542027 
541651 
541076 
540600 
640125 
539651 
539177 
638703 
538230 
637768 
.537286 
636814 
636342 
635871 
536401 
534931 
634461 
533992 
633524 
633056 
.532687 
632120 
531653 
531186 
530720 
630254 
529789 
629324 
528859 
628395 
,627932 
527468 
527005 
526643 
526081 
525619 
525158 
524697 
524237 
523777 
,523317 
622868 
522399 
521941 
521483 
621026 
620668 
620111 
519655 
519199 
,518743 
618288 
517833 
517379 
516925 
516471 
516018 
515565 
516113 
614661 


27664 
27692 
27620 
27648 
27676 
27704 
27731 
27769 
27787 
27816 
27843 
27871 
27899 
27927 
27956 
27983 
28011 
28039 
28067 
28095 
28123 
28150 
28178 
28206 
28234 
28262 
28290 
28318 
28346 
28374 
28402 
28429 
28457 
28485 
28513 
28641 
28569 
28597 
28626 
28652 
28680 
28708 
28736 
28764 
28792 
2882U 
28847 
j  28875 
28903 
28931 
2895*.. 
2898'*/ 
29015 
29042 
290/0 
29098 
29126 
29154 
29182 
29201' 
29247 


Tang.   li  N.  cos.  N.sine. 


96126 
96118 
96110 
96102 
96094 
96086 
96078 
96070 
96062 
96064 
96046 
96037 
96029 
96021 
96013 
96006 
95997 
95989 
95981 
95972 
95964 
95956 
96948 
95940 
96931 
95923 
95916 
95907 
95898 
95890 
95882 
96874 
95865 
96867 
95849 
96841 
95832 
95824 
95816 
y5807 
96799 
96791 
95782 
95774 
95766 
95767 
96749 
95740 
95732 
95724 
95715 
95707 
95698 
95690 
95681 
95673 
95664 
95056 
95647 
95639 
95630 


73  Degrees. 


38 


Log.  Sines  and  Tangents.  (17°)  Natural  Sines. 


TABLE  IL 


Sine. 


1 

2 
3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

40 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 


0(9.465935 
466348 
466761 
467173 
467585 
467996 
468407 
468817 
469227 
469637 
470046 

9.470455 
470863 
471271 
471679 
472086 
472492 
472898 
473304 
473710 
474115 

9.474619 
474923 
475327 
475730 
476133 
476536 
476938 
477340 
477741 
478142 

9.478542 
478942 
479342 
479741 
480140 
480539 
480937 
481334 
481731 
482128 

9.482525 
482921 
483316 
483712 
484107 
484501 
484895 
485289 
485682 
480075 

9.486467 
486860 
487251 
487643 
488034 
488424 
488814 
489204 
489593 
489982 


D.  10" 


Cosine. 


68.8 

68.8 

68.7 

68.6 

68.5 

68.5 

68.4 

68.3 

68.3 

68.2 

68.1 

68.0 

68.0 

67.9 

67.8 

67.8 

67.7 

67.6 

67.6 

67.5 

67.4 

67.4 

67.3 

67.2 

67.2 

67.1 

67.0 

66.9 

66.9 

66.8 

66.7 

66.7 

66.6 

66.5 

66.5 

66.4 

66.3 

66.3 

66.2 

66.1 

66.1 

66.0 

65.9 

65.9 

65.8 

65.7 

65.7 

66.6 

65.5 

65.5 

65.4 

66.3 

65.3 

65.2 

65.1 

65.1 

66.0 

65.0 

64.9 

64.8 


Cosine. 


9.980596 
980558 
980519 
980480 
980442 
980403 
980364 
980325 
980286 
980247 
980208 
.980169 
980130 
980091 
980052 
980012 
979973 
979934 
979895 
979855 
979816 
97977a 
979737 
979697 
979668 
979618 
979579 
979539 
979499 
979459 
979420 

9.979380 
979340 
979300 
979260 
979220 
979180 
979140 
979100 
979059 
979019 

9.978979 
978939 
978898 
978858 
978817 
978777 
978736 
978696 
978655 
978615 

9,978574 
978533 
978493 
978462 
978411 
978370 
978329 
978288 
978247 
978206 


D.  10'^ 


Sine. 


6.4 
6.4 
6.5 
6.5 
6.5 
6.5 
6.5 
6.5 
6.5 
6.5 
6.5 
6.5 
6.5 
6.5 
6.5 
6.5 


Tang.  ID.  10 


6.6 
6.6 
6.6 
6.6 
6.6 
6.6 
6.6 
6.6 
6.6 
6.6 
6.6 
6.6 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.8 
6.8 
6.8 
6.8 
6.8 
6.8 
6.8 
6.8 
6.8 
6.8 
6.8 
6.8 


9.485339 
485791 
486242 
486693 
487143 
487593 
488043 
488492 
488941 
489390 
489838 
9.490286 
490733 
491180 
491627 
492073 
492519 
492965 
493410 
493854 
494299 
9.494743 
495186 
495630 
496073 
496515 
496957 
497399 
497841 
468282 
498722 
9.499163 
499603 
500042 
500481 
500920 
501359 
501797 
502235 
502672 
503109 
503546 
503982 
504418 
504854 
505289 
505724 
506159 
5U6593 
507027 
607460 
507893 
508326 
508759 
509191 
509622 
510054 
510486 
610916 
511346 
511776 


Co  tang. 


3 
75.2 
76.1 
75.1 
75.0 
74.9 
74.9 
74.8 
74.7 
74.7 
74.6 
74.6 
74.5 
74.4 
74.4 
74.3 
74.3 
74.2 
74.1 
74.0 
74.0 
74.0 
73.9 
73.8 
73.7 
73.7 
73.6 
73.6 
73.5 
73.4 
73.4 
73.3 
73.3 
73.2 
73.1 
73.1 
73.0 
73.0 
72.9 
72.8 
72.8 
72.7 
72.7 
72.6 
72.5 
72.5 
72.4 
72.4 
72.3 
72.2 
72.2 
72.1 
72.1 
72.0 
71.9 
71.9 
71.8 
71.8 
71.7 
71.6 


Cotang.  IN.  sine 


10.514661 
514209 
513758 
513307 
512867 
612407 
511967 
511508 
511059 
510610 
610162 
10.509714 
509267 
508820 
508373 
607927 
507481 
507035 
506590  i 
506146 
505701 
10.505257 
504814 
604370 
503927 
603485 
503043 
502601 
502159 
601718 
501278 
10.500837 
500397 
499968 
499619 
499080 
498041 
498203 
497765 
497328 
496891 
10.496454 
496018 
495682 
496146 
494711 
494276 
493841 1 
493407 
492973  I 
492540 1 
10.492107; 
491674 
491241 
490809 
490378 
489946 
489515 
489084 
488654 
488224 


1:29237 
I  i 29265 
1^29293 
29321 
29348 
29376 
29404 
29432 


95630 
95622 
95613 
95606 
95596 
95588 
95579 
95671 


29460196562 


!2948 
129615 
! 29543 
129571 
I  29599 
29620 
! 29654 
j 29682 
129710 
i  29737 
j  29765 
1 29793 
I  29821 


95654 
'95545 
95536 
95528 
95519 
95611 
95502 
95493 
95485 
95476 
95467 
95459 
95450 


2984995441 
29876  95433 
29904  95424 


Tang. 


29932 
29960 
2998 
130015 
I  30043 
I  30071 
1 30098 
30126 
30154 
30182 
30209 
30237 
30265 
30292 
30320 
30348 
30376 
30403 
30431 
30459 
30486 
30514 
30542 
30570 
3059  ( 
30625 
30053 
30080 
30708 


30703 
30791 
30819 
30840 
30874 
30902 


j  N.  COS.  N.pine, 


96416 
95407 
95398 
95389 
95380 
95372 
95303 
95354 
95345 
95337 
95328 
95319 
95310 
95301 
95293 
95284 
95276 
95266 
95257 
95248 
95240 
95231 
95222 
95213 
95204 
95195 
95186 
95177 
95168 


30730  95159 


95160 
95142 
95133 
95124 
96115 
95100 


60 
69 
68 
67 
66 
65 
54 
63 
62 
51 
60 
49 
48 
47 
46 
45 
44 
43 
42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

26 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

U 

10 
9 
8 
7 
6 
6 
4 
3 
2 
1 
0 


7?  Degrees. 


TABLE  II. 


Log.  Sines  and  Tangents.    (18°)    Natural  Sines. 


39 


Sine. 


D.  10' 


9.489982 
490371 
490759 
491147 
491535 
491922 
492308 
492695 
493081 
493466 
493861 

9.494236 
494621 
495005 
495388 
495772 
496154 
496537 
496919 
497301 
497682 
498064 
498444 
498826 
499204 
499584 
499963 
500342 
500721 
501099 
501476 

9.501854 
502231 
502607 
502984 
503360 
503735 
504110 
504485 
504860 
505234 

9.505608 
505981 
506354 
506727 
507099 
507471 
507843 
508214 
508586 
508956 

9.509326 
509696 
510065 
510434 
510803 
511172 
511540 
511907 
512275 
512642 


Cosine. 


64.8 
64.8 
64.7 
64.6 
64.6 
64.6 
64.4 
64.4 
64.3 
64.2 
64.2 
64.1 
64.1 
64.0 
63.9 
63.9 
63.8 
63.7 
63.7 
63.6 
63.6 
63.5 
63.4 
63.4 
63.3 
63.2 
63.2 
63.1 
63.1 
63.0 
62.9 
62.9 
62.8 
62.8 
62.7 
62.6 
62.6 
62.5 
62.5 
62.4 
62.3 
62.3 
62.2 
62.2 
62.1 
62.0 
62.0 
61.9 
61.9 
61.8 
61.8 
61.7 
61.6 
61.6 
61.5 
61.5 
61.4 
61.3 
61.3 
61.2 


9. 


Cosine. 


D.  10' 


9. 


978206 
978165 
978124 
978083 
978042 
978001 
977959 
977918 
977877 
977835 
977794 
977752 
977711 
977669 
977628 
977686 
977544 
977503 
977461 
977419 
977377 
977335 
977293 
977251 
977209 
977167 
977126 
977083 
977041 
976999 
976957 
976914 
976872 
976830 
976787 
976745 
976702 
976660 
976617 
976574 
976532 
976489 
976446 
976404 
976361 
976318 
976275 
976232 
976189 
976146 
976103 
,976060 
976017 
976974 
975930 
975887 
976844 
975800 
975767 
975714 
976670 


Sine. 


6.8 
6.8 
6.8 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
7.0 
7.0 
7.0 
7.0 
7.0 
7.0 
7.0 
7.0 
7.0 
7.0 
7.0 
7.0 
7.0 
7.0 
7.0 
7.0 
7.1 
7.1 
7.1 
7.1 
7.1 
7.1 
7.1 
7.1 
7.1 
7.1 
7.1 
7.1 
7.1 
7.1 
7.1 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 


Tang. 


D.  10" 


611776 
512206 
612636 
613064 
513493 
513921 
514349 
514777 
616204 
515631 
616067 
616484 
616910 
517335 
617761 
518185 
518610 
519034 
519458 
519882 
520306 

9.520728 
621161 
621573 
521995 
522417 
522838 
523259 
523680 
624100 
524620 

9.524939 
525369 
525778 
526197 
626615 
627033 
527451 
527868 
528285 
528702 

9.629119 
629636 
629960 
530366 
530781 
531196 
631611 
532025 
632439 
632853 
633266 
533679 
534092 
534504 
534916 
636328 
636739 
636160 
536561 
636972 


Cotang. 


71.6 
71.6 
71.5 
71.4 
71.4 
71.3 
71.3 
71.2 
71.2 
71.1 
71.0 
71.0 
70.9 
70.9 
70.8 
70.8 
70.7 
70.6 
70.6 
70.5 
70.6 
70.4 
70.3 
70.3 
70.3 
70.2 
70.2 
70.1 
70.1 
70.0 
69.9 
69.9 
69.8 
69.8 
69.7 
69.7 
69.6 
69.6 
69.6 
69.5 
69.4 
69.3 
69.3 
69.3 
69.2 
69.1 
69.1 
69.0 
69,0 
68.9 
68.9 
68.8 
68.8 
68.7 
68.7 
68.6 
68.6 
68.6 
68.6 
68.4 


Cotang.     I N.  sine.  N.  cos 


10 


10 


10 


10 


10 


10 


.488224  i 
487794 ! 
487366  I 
486936 
486607 
486079 
486651 
485223 
484796 
484369 
483943 

.483516 
483090 
482665 
482239 
481815 
481390 
480966 
480542 
480118 
479695 

.479272 
478849 
478427 
478005 
477583 
477162 
476741 
476320 
475900 
476480 

.475061 
474641 
474222 
473803 
473386 
472967 
472649 
472132 
471716 
471298 

.470881 
470466 
470050 
469634 
469219 
468804 
468389 
467975 
467661 
467147 

.466734 
466321 
466908 
466496 
465084 
464672 
464261 
463850 
463439 
463028 


Tang. 


30902 
30929 
30957 
30985 
31012 
31040 
31068 
31095 
31123 
31151 
31178 
31206 
31233 
31261 
31289 
31316 
31344 


31372  94952 


31399 
31427 
31464 
31482 
31510 
31537 
31565 
31593 
31620 
31648 
31675 
31703 
31730 
31768 
31786 
31813 
31841 
31868 
31896 
31923 
31961 
31979 
32006 
32034 
32061 
32089 
32116 
32144 
32171 
32199 
32227 


96106 
96097 
96088 
95079 
95070 
95061 
95052 
96043 
95033 
95024 
95016 
95006 
94997 
94988 
94979 
94970 
94961 


94943 
94933 
94924 
94916 
94906 
94897 
94888 
94878 
94869 
94860 
94851 
94842 
94832 
94823 
94814 
94805 
94795 
94786 
94777 
94768 
94768 
94749 
94740 
94730 
94721 
94712 
94702 
94693 
94684 
94674 
94666 
32250  94656 
32282  94646 


32309 
32337 
32364 
32392 
32419 
32447 
32474 
32502 
32629 
32557 


94637 
94627 
94618 
94609 
94599 
94590 
94580 
94571 
94561 
94652 
N.  cos.  N-sine. 


71  Degrees. 


40 


Log.  Sines  and  Tangents.  (19°)  Natural  Sines. 


TABLE  IL 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 


Sine. 


9.512642 
513009 
513375 
513741 
514107 
514472 
514837 
515202 
515566 
515930 
516294 
9.516657 
517020 
517382 
517745 
518107 
518468 
518829 
519190 
519551 
519911 
9.520271 
520631 
520990 
521349 
521707 
522066 
522424 
522781 
523138 
523495 
9.523852 
524208 
524664 
524920 
525275 
625680 
525984 
626339 
626693 
527046 
9.527400 
627753 
628106 
628468 
528810 
529161 
629613 
529864 
530215 
530565 
530915 
531265 
531614 
531963 
532312 
532661 
533009 
533357 
533704 
534052 


D.  lu' 


61.2 
61.1 
61.1 
61.0 
60.9 
60.9 
60.8 
60.8 
60.7 
60.7 
60.6 
60.5 
60.5 
60.4 
60.4 
60.3 
60.3 
60.2 
60.1 
60.1 
60.0 
60.0 
59.9 
59.9 
59.8 
59.8 
59.7 
69.6 
59.6 
59.5 
59.6 
4 
59.4 
59.3 
59.3 
69.2 
69.1 
59.1 
59.0 
59.0 
58.9 
58.9 
58.8 
58.8 
58.7 
58  7 
58.6 
58.6 
58.5 


Losme. 


Cosine. 


58.4 
68-3 
68.2 
58-2 
58.1 
58.1 
58.0 
58.0 
67.9 


975670 
975627 
975583 
975539 
97.5496 
975452 
975408 
975365 
975321 
975277 
975233 
9.975189 
975145 
975101 
975057 
975013 
974969 
974925 
974880 
974836 
974792 
9.974748 
974703 
974659 
974614 
974570 
974525 
974481 
974436 
974391 
974347 
,974302 
974267 
974212 
974167 
974122 
974077 
974032 
973987 
973942 
973897 
9.973852 
973807 
973761 
973716 
973671 
973625 
973580 
973535 
973489 
973444 
19.973398 
973352 
973307 
973261 
973215 
973169 
973124 
973078 
973032 
972986 


D.  lu' 


Sine. 


7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.4 

7.4 

7.4 

7.4 

7.4 

7.4 

7.4 

7.4 

7.4 

7.4 

7.4 

7.4 

7.4 

7.4 

7.5 

7.5 

7.5 

7.6 

7.6 

7.6 

7.6 

7.6 

7.5 

7.6 

7.6 

7.6 

7.6 

7.6 

7.6 

7.6 

7.6 

7.6 

7.6 

7.6 

7.6 

7.6 

7.6 

7.6 

7.6 

7.6 

7.6 

7.6 

7.6 

7.7 


i'ang.  (l>^</'j  Cotang.  ;  |N.  sine.)  N .  cos. j 


,536972 
537382 
537792 
538202 
538611 
539020 
539429 
539837 
540245 
540653 
541061 
9.541468 
541875 
542281 
542688 
543094 
543499 
543905 
544310 
644715 
645119 
9.546524 
546928 
646331 
646735 
547138 
547540 
547943 
548345 
648747 
649149 
9.549550 
549951 
550352 
660752 
651152 
551662 
651962 
652351 
662760 
653149 
9.563548 
653946 
664344 
554741 
655139 
555536 
666933 
656329 
566725 
657121 
9.557517 
557913 
558308 
558702 
659097 
559491 
559885 
560279 
560673 
561066 


Cotang. 


68.4 

68.3 

68.3 

68.2 

68.2 

68.1 

68.1 

68.0 

68.0 

67.9 

67.9 

67.8 

67.8 

67.7 

67.7 

67.6 

67.6 

67.5 

67.5 

67.4 

67.4 

67.3 

67.3 

67.2 

67.2 

67.1 

67.1 

67.0 

67.0 

66.9 

66.9 

66.8 

66.8 

66.7 

66.7 

66.6 

66.6 

66.6 

66.5 

66.5 

66.4 

66.4 

66.3 

66.3 

66.2 

66.2 

66.1 

66.1 

66.0 

66.0 

66.9 

66.9 

65.9 

65.8 

65.8 

65.7 

65.7 

65.6 

65.6 

65.5 


480980  32694 
460571  !  32722 
460163  132749 


3277' 


469755 

459347  ;i32804j 

458939  13283-2 
10.458532  132859 

458125  1 32887 

457719 

457312 

456906 

456601 

456095133024 

465690  133051 

455285  133079 

454881  [33106 
10.454476 

454072 

453669 

453265 

452862 

452460 

452057 

451655 

451253 

450851 
10.450450 
450049 
449648 
449248 
448848 
448448 
448048 
447649 ' 
447250  133627 
446851 
10.4464521 1 33682 
446054133710 
445656  '33737 
445259  '33764 
444861  33792 
444464  33819 
444067  ,'33846 
443671  '33874 
443275  33901 
442879  33929 


32914 
32942 
32969 
'  3299 


33134 
33161 
33189 
33216 
33244 
33271 
33298 
33326 
33353 
33381 
33408 
33436 
33463 
33490 
33518 
33545 
33573 
33600 


33655  94167 
94157 
94147 


10.442483 
442087 
441692 
441298 
440903 
440509 
440115 
439721 
439327 
438934 


33983 
34011 
34038 
34065 
34093 
34120 
34147 
34175 
34202 


Tang. 


N.  COS.  .V.sine, 


94552 

94542 

94533 

94523 

94514 

94504 

94495 

94485 

94476 

94466 

94457 

94447 

94438 

94428 

94418 

94409 

94399 

94390 

94380 

94370 

94361 

94351 

94342 

94332 

94322 

94313 

94303 

94293 

94284 

94274 

94264 

94254 

94245 

94236 

94225 

94215 

94206 

94196 

94186 

94176 


94137 
94127 
94118 
94108 
94098 
94088 
94078 
94068 


33956  94058 


94049 
94039 
94029 
94019 
94009 
93999 
93989 
93979 
93969 


60 

59 

58 

57 

56 

55 

64 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

16 

14 

13 

12 

11 

10 
9 
8 
7 
6 
6 
4 
3 
2 
1 
0 


70  Degrees. 


TABLE  II. 


Log.  Sines  and  Tangents.    (20°)    Natural  Sines. 


41 


0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


Sine. 

1.534052 
534399 
534745 
535092 
535438 
635783 
536129 
536474 
536818 
537163 
537507 

1.537851 
538194 
538538 
538880 
539223 
539565 
539907 
540249 
540590 
540931 

1.541272 
541613 
541953 
542293 
542632 
642971 
543310 
543649 
543987 
544326 

1.544663 
645000 
645338 
545674 
546011 
546347 
546683 
547019 
547354 
647689 

.548024 
548359 
648693 
549027 
549360 
549693 
550026 
550359 
650692 
651024 

.651356 
551687 
552018 
552349 
552680 
653010 
553341 
553670 
564000 
554329 


Cosine. 


D.  10^' 

57.8 
67.7 
57.7 
67-7 
57.6 
57.6 
57.5 
57.4 
67.4 
67.3 
67.3 
67.2 
57.2 
57.1 
67.1 
67.0 
67.0 
56.9 
56.9 
56.8 
66.8 
56.7 
66.7 
66.6 
56.6 
56.5 
66.6 
66.4 
56.4 
56.3 
66.3 
66.2 
66.2 
56.1 
56.1 
66.0 
56.0 
55.9 
55.9 
65.8 
65.8 
65.7 
65.7 
56.6 
55.6 
55.6 
65.5 
55.4 
65.4 
56.3 
65.3 
65.2 
65.2 
65.2 
65.1 
55.1 
55.0 
65.0 
54.9 
54.9 


Cosine. 


9.972986 
972940 
972894 
972848 
972802 
972755 
97270y 
972663 
972617 
972570 
972524 

9.972478 
972431 
972385 
972338 
972291 
972245 
972198 
972161 
972105 
972058 

9.972011 
971964 
971917 
971870 
971823 
971776 
971729 
971682 
971635 
971588 

9.971540 
971493 
971446 
971398 
971351 
971303 
971256 
971208 
971161 
971113 

9.971066 
971018 
970970 
970922 
970874 
970827 
970779 
970731 
970683 
970635 

9.970586 
970638 
970490 
970442 
970394 
970345 
970297 
970249 
970200 
970162 


Sine. 


D.  10" 


7.7 
7.7 
7.7 
7.7 
7.7 
7.7 
7.7 
7.7 
7.7 
7.7 
7.7 
7.7 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7,9 
7.9 
7,9 
7.9 
7.9 
7.9 
7.9 
7.9 
7.9 
7.9 
7.9 
7.9 
7.9 
7.9 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.1 
8.1 
8.1 
8.1 


Tang. 


9.661066 
661459 
561851 
562244 
562636 
563028 
563419 
563811 
664202 
564592 
564983 

9.566373 
566763 
566153 
566542 
566932 
567320 
667709 
668098 
568486 
568873 

9.669261 
569648 
670035 
670422 
570809 
671195 
571581 
571967 
572352 
572738 

9.673123 
673507 
673892 
574276 
574660 
575044 
575427 
675810 
576193 
676576 
576968 
577341 
577723 
578104 
578486 
578867 
579248 
579629 
680009 
680389 

9.580769 
681149 
681528 
581907 
682286 
682666 
683043 
683422 
583800 
684177 


Cotang. 
Degrees. 


D.  10" 


65.5 
66.4 
66.4 
65.3 
65.3 
65.3 
65.2 
65.2 
65.1 
66.1 
65.0 
65.0 
64.9 
64.9 
64.9 
64.8 
64.8 
64.7 
64.7 
64.6 
64.6 
64.6 
64.6 
64.6 
64.4 
64.4 
64.3 
64.3 
64.2 
64.2 
64.2 
64.1 
64.1 
64.0 
64.0 
63.9 
63.9 
63.9 
63.8 
63.8 
63.7 
63.7 
63.6 
63.6 
63.6 
63.6 
63.6 
63.4 
63.4 
63.4 
63.3 
63.3 
63.2 
63.2 
63.2 
63.1 
63.1 
63.0 
63.0 
62.9 


Cotang.     I  N.  sine.  N.  cos . 


10.438934 
438541 
438149 
437756 
437364 
436972 
436581 
436189 
435798 
435408 
435017 

10.434627 
434237 
433847 
433458 
433068 
432680 
432291 
431902 
431614 
431127 

10.430739 
430352 
429965 
429578 
429191 
428805 
428419 
428033 
427648 
427262 

10.426877 
426493 
426108 
425724 
425340 
424956 
424573 
424190 
423807 
423424 

10.423041 
422669 
422277 
421896 
421514 
421133 
420762 
420371 
419991 
419611 

10.419231 
418861 
418472 
418093 
417714 
417335 
416957 
416578 
416200 
415823 


34202 
34229 
34257 

34284 
34311 
34339 
34366 
34393 
34421 
34448 
34476 
34503 
34530 
34567 
34584 
34612 
34639 
34666 
34694 
34721 
34748 
34776 
34803 
34830 
34857 
34884 
34912 
34939 
34966 
34993 
35021 
3504b 
36076 
1135102 
!  35130 
!  I  35167 
'  35184 
35211 
35239 
36266 
36293 
35320 
35347 
35376 
3540i 
35429 
35456 
35484 
35511 
35538 
35565 
35592 
35619 
35647 
35674 
35701 
35728 
35755 
35782 
35810 
35837 


Tang. 


93969 
93969 
93949 
93939 
93929 
93919 
93909 
93899 
93889 
93879 
93869 
93859 
93849 
93839 
93829 
93819 
93809 
93799 
93789 
93779 
93769 
93759 
93748 
93738 
93728 
93718 
93708 
93698 
93688 
93677 
93667 
93667 
93647 
93637 
93626 
93616 
93606 
93596 
93686 
y3576 
93565 
93555 
9o544 
93534 
93524 
93614 
93503 
93493 
93483 
93472 
93462 
93452 
93441 
93431 
93420 
93410 
93400 
93389 
93379 
93368 
93368 
N.  cos.  N.sine. 


42 


Log.  Sines  and  Tangents.    (21°)    Natural  Sines. 


TABLE  IL 


Sine. 


9.554329 
554658 
554987 
555315 
555643 
555971 
556299 
556626 
556953 
557280 
557606 
557932 
558258 
558583 
558909 
559234 
659558 
559883 
560207 
560531 
560855 

9.561178 
561501 
561824 
562146 
562468 
562790 
563112 
563433 
563755 
564075 
564396 
564716 
565036 
565356 
565676 
565995 
566314 
566632 
566951 
567269 

9.567587 
567904 
568222 
568539 
568856 
569172 
569488 
569804 
570120 
570435 

9.570751 
571066 
571380 
571695 
572009 
572323 
572636 
572950 
573263 
573575 


D.  10"!  Cosine.  iD.  10" 


Cosine. 


54.8 
54.8 
54.7 
54.7 
54.6 
54.6 
54.5 
54.5 
54.4 
54.4 
54.3 
54.3 
54.3 
54.2 
54.2 
54.1 
54.1 
54.0 
54.0 
53.9 
53.9 
53.8 
53.8 
53.7 
53.7 
53.6 
53.6 
53.6 
53.5 
53.5 
53.4 
53.4 
53.3 
53.3 
53.2 
53.2 
53.1 
53.1 
53.1 
53.0 
53.0 
52.9 
52.9 
52.8 
52.8 
52.8 
52.7 
52.7 
52.6 
52.6 
52.5 
52.5 
52.4 
52.4 
52.3 
52.3 
52.3 
52.2 
52.2 
52.1 


.970152 
970103 
970055 
970006 
969957 
969909 
969860 
969811 
969762 
969714 
969665 
.969616 
969567 
969518 
969469 
969420 
969370 
969321 
969272 
969223 
969173 
.969124 
969075 
969025 
968976 
968926 
968877 
968827 
968777 
968728 
968678 
968628 
968578 
968528 
968479 
968429 
968379 
968329 
968278 
968228 
968178 
968128 
968078 
968027 
967977 
967927 
967876 
967826 
967775 
967725 
967674 
9G7624 
967573 
967522 
967471 
967421 
967370 
967319 
967268 
967217 
967166 
Sine. 


2 
2 
2 
2 
2 
8.2 
8.2 
8.2 
8.2 
8.2 
8.2 
8.2 
8.2 
8.2 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.4 
8.5 
8.5 
8.5 
8.5 
8.5 
8.5 
8.5 


Tang. 


9.584177 
584555 
584932 
585309 
585686 
586062 
586439 
586815 
587190 
587566 
587941 

9.588316 
588691 
589066 
589440 
589814 
590188 
590562 
590935 
591308 
591681 

9.592064 
592426 
592798 
593170 
593542 
593914 
594285 
594656 
596027 
595398 

9.595768 
596138 
696508 
596878 
597247 
597616 
597985 
598354 
598722 
599091 

9.599469 
699827 
600194 
600562 
600929 
601296 
601662 
602029 
602395 
602761 

9.603127 
603493 
603858 
604223 
604588 
604963 
605317 
605682 
(.06046 
606410 


D.  10"|  Cotang.  !  ;N  .sine.  N.  coa. 


62.9 

62.9 

62.8 

62.8 

62.7 

62.7 

62.7 

62.6 

62.6 

62 

62.5 

62.5 

62.4 

62.4 

62.3 

62.3 

62.3 

62.2 

62.2 

62.2 

62.1 

62.1 

62.0 

62.0 

61.9 

61.9 

61.8 

61.8 

61.8 

61.7 

61.7 

61.7 

61.6 

61.6 

61.6 

61.5 

61.5 

61.6 

61.4 

61.4 

61.3 

61.3 

61.3 

61.2 


61.2 
61.1 
61.1 
61.1 
61.0 
61.0 
61.0 
60.9 
60.9 
60.9 
60.8 
60.8 
60.7 
60.7 
60.7 
60.6 


!lO. 415823 
415445 
415068 
414691 
414314 
413938 
413561 
413185 
412810 
412434 
412059 

10.411684 
411309 
410934 
410560 
410186 
409812 
40943b 
409065 
408692 
408319 

10.407946 
407574 
407202 
406829 
406458 
406086 
405715 
405344 
404973 
404602 

10.404232 
403862 
403492 
403122 
402753 
402384 
402015 
401646 
401278 
400909 

10.400541 
400173 
399806 
399438 
399071 
398704 
398338 
397971 
397605 
397239 

10.396873 
396507 
396142 
395777 
395412 
395047 
394683 
394318 
393954 
393590 


135837 
{35864 
135891 

I  i 35918 

I I  35945 
35973 

: 36000 
M 36027 

I  36054 
{'36081 

{36108 
I  36135 
36162 
36190 
1  36217 
h 36244 

: 36271 
,:  36298 
{'36325 
{j  36352 
1136379 

'36406 
i  1 36434 

136461 
:{364S8 
36515 
:  36542 
:  36569 

36596 
jj  36623 

'  36650 
1 1 36677 
!  1 36704 
:  136731 

I  36758 

30785 

36812 

i 36839 

3686 


36894  92945 


Cotang. 
Degrees. 


36921 
36948 
36975 
37002 
37029 
3705b 
37083 
37110 
37137 
37164 
37191 
37218 
37245 
37272 
37299 
37326 
37353 
37380 
37407 
37434 
37461 
Tang.   'i  N.  cos. 


93358 
93348 
93337 
93327 
93316 
93306 
93295 
93285 
93274 
93264 
93253 
93243 
93232 
93222 
93211 
93201 
93190 
93180 
IJ3169 
93159 
93148 
93137 
93127 
93116 
93106 
93095 
93084 
93074 
93063 
93052 
93042 
93031 
93020 
93010 
92999 
92988 
92978 
92967 
92956 


92936 
92926 
92913 
92902 
J2892 
92881 
J2870 
92859 
J2849 
J2838 
i2827 
92816 
92805 
92794 
92784 
92773 
92762 
92751 
92740 
92729 
92718 


N.sine, 


TABLE  II. 


Log.  Sines  and  Tangents.    (22°)    Natural  Sines. 


43 


Sine. 


D.  10" 


.573575 

573888 
574200 
574512 
574824 
575136 
575447 
575758 
57G069 
576379 
576689 

.576999 
577309 
577618 
577927 
578236 
578545 
578853 
579162 
579470 
579777 

.580085 
580392 
580699 
581005 
581312 
581618 
581924 
582229 
582535 
582840 

.583145 
583449 
583754 
584058 
584361 
584665 
584968 
585272 
585574 
585877 

.586179 
586482 
586783 
587085 
587386 
587688 
587989 
588289 
588590 
588890 

.589190 
589489 
589789 
590088 
590387 
590686 
590984 
591282 
591580 
591878 


Cosine. 


52.1 
52.0 
52.0 
51.9 
51.9 
51.9 
51.8 
51,8 
51.7 
51.7 
61.6 
51.6 
51.6 
51.5 
51.5 
51.4 
51,4 
51,3 
51.3 
51.3 
51.2 
51.2 
51,1 
51.1 
51,1 
51,0 
51.0 
50.9 
50.9 
50.9 
50,8 
50,8 
50,7 
60.7 
60.6 
50.6 
50.6 
50.5 
50.6 
60.4 
50.4 
60.3 
50.3 
50.3 
50.2 
60.2 
50.1 
50.1 
50.1 
50.0 
60.0 
49.9 
49,9 
49.9 
49.8 
49.8 
49,7 
49.7 
49,7 
49.6 


Cosine.     D.  10' 


.967166 
967116 
967064 
967013 
966961 
966910 
966859 
966808 
966756 
966705 
966653 

.966602 
966650 
966499 
966447 
966395 
966344 
966292 
966240 
966188 
966136 

.966085 
966033 
966981 
965928 
966876 
965824 
966772 
966720 
965668 
965615 

.965663 
965611 
966468 
966406 
966363 
965301 
966248 
965195 
966143 
965090 

•965037 
964984 
964931 
964879 
964826 
964773 
964719 
964666 
964613 
964560 

.964507 
964454 
964400 
964347 
964294 
964240 
964187 
964133 
964080 
964026 


Sine. 


8.5 

8,6 
8.6 
8,5 
8.5 
8.5 
8.6 
8.5 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8,7 
8.7 
8.7 
8.7 
8.7 
8.8 
8.8 
8.8 
8.8 
8.8 
8.8 
8,8 
8,8 
8.8 
8.8 
8.8 
8.8 
8.8 
8.9 
8.9 
8.9 


8.9 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 


Tang. 


D.  10' 


9.606410 
606773 
607137 
607600 
607863 
608225 
608588 
608950 
609312 
609674 
610036 
610397 
610759 
611120 
611480 
611841 
612201 
612561 
612921 
613281 
613641 

9.614000 
614359 
614718 
616077 
615435 
615793 
616151 
616509 
616867 
617224 

9.617682 
617939 
618295 
618652 
619008 
619364 
619721 
620076 
620432 
620787 

9.621142 
621497 
621852 
622207 
622661 
622915 
623269 
623623 
623976 
624330 
624683 
625036 
625388 
626741 
626093 
626445 
626797 
627149 
627501 
627852 


Cotang. 


Cotang.  I  N.  sine.  N.  cos. 


10.393590 
393227 
392863 
392500 
392137 
391776 
391412 
391050 
390688 
390326 
389964 

10.389603 
389241 
388880 
388620 
388169 
387799 
387439 
387079 
386719 
386369 

10-386000 
386641 
386282 
384923 
384565 
384207 
383849 
383491 
383133 
382776 

10-382418 
382061 
381705 
381348 
380992 
380636 
380279 
379924 
379568 
379213 

10-378858 
378503 
378148 
377793 
377439 
377085 
376731 
376377 
376024 
376670 

10.376317 
374964 
374612 
374259 
373907 
373655 
373203 
372851 
372499 
372148 
Tang. 


37461 

37488 
37516 
37642 
j  37569 
1 37595 
! 37622 
1 37649 
1 37676 
j  37703 
37730 
1 37757 
37784 
37811 
37838 
37865 
37892 
37919 
37946 
37973 
37999 
38026 
38053 
38080 
38107 
38134 
38161 
38188 
38216 
38241 
38268 
38296 
38322 


92718 
92707 
92697 
92686 
92675 
926()4 
92663 
92642 
92631 
92620 
92609 
92598 
92587 
92576 
92665 
92554 
92543 
92532 
92521 
92610 
92499 
92488 
92477 
92466 
92455 
92444 
92432 
92421 
92410 
92399 
92388 
92377 
92366 


38349  92365 


38376 
38403 
38430 


92343 
92332 
92321 


38456  92310 
38483(92299 


38610 
38537 
38564 
38591 
38617 
38644 
38671 
38698 
38726 
38752 


92287 
92276 
92266 
92254 
92243 
92231 
92220 
92209 
92198 
92186 


38778  !921 76 


38805 
38832 
38859 


92164 
92152 
92141 


3888692130 
38912  92119 
3893992107 
38966192096 


38993 
39020 
39046 
39073 


92085 
92073 
92062 
92050 
N.  cos.j  N.sine. 


67  Degrees. 


44 


Log.  Sines  and  Tangents.    (23°)    Natural  Sines.  TABLE  II. 


0 

1 

2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
46 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


Sine. 


9.591878 
59-2176 
592473 
592770 
5930S7 
593363 
593659 
593955 
594251 
594547 
594842 

9.595137 
595432 
695727 
596021 
596315 
596609 
596903 
597196 
597490 
597783 

9.598075 
598368 
598660 
598952 
599244 
599536 
599827 
600118 
600409 
600700 

9.600990 
601280 
601570 
601860 
602150 
602439 
602728 
603017 
603305 
603594 

9.603882 
604170 
604457 
604745 
606032 
005319 
605606 
605892 
606179 
606465 
606751 
607036 
607322 
607607 
607892 
608177 
608461 
608745 
609029 
609313 


Cosine. 


D.  lu"|  Cosine. 

1.964026 
963972 
963919 
963865 
963811 
963757 
963704 
963650 
963596 
963542 
963488 

'.963434 
963379 
963325 
963271 
963217 
963163 
963108 
963054 
962999 
962945 
.962890 
962836 
962781 
962727 
962672 
962617 
962562 
962508 
962453 
962398 

.962343 
962288 
962233 
962178 
962123 
962067 
962012 
961957 
961902 
961846 

.961791 
961735 
961680 
961624 
961569 
961513 
961458 
961402 
961346 
961290 

.961236 
961179 
961123 
961067 
961011 
960956 
960899 
960843 
960786 
960730 


49.6 

49.5 

49.5 

49.5 

49.4 

49 

49 

49 

49 

49 

49 

49.1 

49.1 

49.1 

49.0 

49.0 

48.9 

48.9 

48.9 

48.8 

48.8 

48.7 

48.7 

48.7 

48.6 

48.6 

48.5 

48.5 

48.5 

48.4 

48.4 

48.4 

48.3 

48.3 

48.2 

48.2 

48.2 

48.1 

48.1 

48.1 

48.0 

48.0 

47.9 

47.9 

47.9 

47  8 

47.8 

47.8 

47.7 

47.7 

47.6 

47.6 

47.6 

47-5 

47-5 

47-4 

47.4 

47.4 

47.3 

47.3 


D.  10' 


Sine. 


8.9 
8.9 
8.9 
9.0 
9.0 
9.0 
9,0 
9.0 
9.0 
9.0 
9.0 
9.0 
9.0 
9.0 
9.0 
9.0 
9.0 
9.1 
9.1 
9.1 


9.1 
9.1 
9.1 
9.1 
9.1 
9.1 
9.1 
9.1 
9.1 
9.1 
9  2 
9.2 
9.2 
9.2 
9.2 
9.2 
9.2 
9.2 
9.2 
9.2 
9.2 
9.2 
9.2 
9.2 
9.3 
9.3 
9.3 
9.3 
9.3 
9.3 
9.3 
9.3 
9.3 
9.3 
9.3 
9.3 
9.3 
9.3 
9.4 
9.4 


Tang.  i).  io' 


9.627852 
628203 
628554 
628905 
629255 
629606 
629956 
630306 
630656 
631005 
631355 

9.631704 
632053 
632401 
632750 
633098 
633447 
633795 
634143 
634490 
634838 

9.635185 
635532 
635879 
636226 
636572 
636919 
637265 
637611 
637966 
638302 

9.638647 
638992 
639337 
639682 
640027 
640371 
640716 
641060 
641404 
641747 

9.642091 
642434 
642777 
643120 
643463 
643806 
644148 
644490 
644832 
645174 
645516 
645857 
646199 
646540 
646881 
647222 
647662 
647903 
648243 
648583 


Cotang. 


58.5 

58.5 

58.5 

58.4 

58.4 

58.3 

58.3 

68.3 

68.3 

58.2 

58.2 

58.2 

58.1 

58.1 

68.1 

58.0 

58.0 

58.0 

67.9 

57.9 

57.9 

57.8 

67.8 

57.8 

57.7 

57.7 

57.7 

67.7 

57.6 

67.6 

57.6 

67.6 

67.6 

67.5 

57.4 

67.4 

67.4 

57.3 

57.3 

67.3 

67.2 

57 

67 

67 

67 

57 

57 

57.0 

57.0 

67.0 

66.9 

56.9 

66.9 

66.9 

66.8 

56.8 

56.8 

56.7 

56.7 

66.7 


CotaQir,   is.  sinc.fN.  cos. 


10.372148 
371797 
371446 
371095 
370745 
370394 
370044 
369694 
369344 
368995; 
368645 

10.368296; 
367947 
367599 
367250 
366902 
366553 !' 
366205 
366857 j 
365610 
365162  I ; 

10.364815  I : 
36446811 
364121 
363774 
363428 
363081 II 
362735! 
362389!! 
362(H4!! 
361698  I 

10.361353  I 
361008; 
360663  ij 
360318  I ! 
359973 
369629  ! 
3592841! 
3589401! 
358596 
358253 

10.357909 
357566 
357223 
356880 
356537 
356194 
355852 
355510 
356168 
354826 

10.354484 
364143 
353801 
353460 
353119 
362778 
352438 
352097 
351757 
351417 


39073 
39100 
3912/ 
39153 
39180 
39207 
39234 
39260 
39287 
39314 
39341 
39367 
39394 
39421 
39448 
39474 
39501 
39528 
39555 
39581 
39608 
39635 
S9661 
39688 
39715 
39741 
39768 


92050 
92039 
92028 
92016 
92005 
91994 
91982 
91971 
91959 
91948 
91936 
91925 
91914 
91902 
91891 
91879 
91868 
91856 
91845 
91833 
91822 
91810 
91799 
91787 
91775 
91764 
9i;62 


39795  91741 


39822 


39876 


39982 
40008 
40035 


91729 


39848  91718 


91706 


39902  91694 
39928  91683 
39955  91671 


91660 
91648 
91636 


Tang. 


40062191625 
4008891613 

40141^1590 


40168 
40195 
40221 
40248 
40275 
40301 
40G28 
40356 
40381 
40408 
40434 
40461 
40488 
40514 
40541 
40567 
40594 
40621 
40647 


915-8 
9l.:66 
915.^5 
91543 
91531 
91519 
G1508 
91496 
91484 
91472 
91461 
91449 
91437 
91425 
91414 
91402 
91390 
91378 
91366 


40674  91355 


N.  COS.  N.sine, 


66  Degrees. 


TABLE  II. 


Log.  Sines  and  Tangents.  (24°)  Natural  Sines. 


45 


0 
1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

16 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32f 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60, 


Sine. 

9,609313 

609597 

609880 

610164 

610447 

610729 

611012 

611294 

611576 

611858 

612140 

9,612421 

612702 

612983 

613284 

613545 

613825 

614105 

614385 

614665 

614944 

9.615223 

615502 

615781 

616080 

616338 

616616 

616894 

617172 

617450 

617727 

9-618004 

618281 

618558 

618834 

619110 

619386 

619662 

619938 

620213 

620488 

9.620763 

621038 

621313 

621687 

621861 

622135 

622409 

622682 

622956 

623229 

9.623612 

623774 

624047 

624319 

624591 

624863 

625135 

625408 

625877 

625948 


D.  lO'l  Cosine. 


47.3 

47.2 

47.2 

47.2 

47.1 

47.1 

47.0 

47.0 

47.0 

46.9 

46.9 

46.9 

46.8 

46.8 

46.7 

46.7 

46.7 

46.6 

46.6 

46.6 

46.5 

46.5 

46.5 

46.4 

46.4 

46.4 

46.3 

46.3 

46.2 

46.2 

46,2 

46,1 

46.1 

46.1 

46.0 

46.0 

46.0 

45,9 

45.9 

45,9 

45.8 

45.8 

45.7 

45.7 

45.7 

45.6 

45,6 

45,6 

45.5 

45.5 

45.5 

45.4 

45.4 

45.4 

45,3 

45.3 

46.3 

45.2 

45.2 

45.2 


D.  10' 


.980730 
960674 
960618 
960561 
960505 
960448 
960392 
960335 
960279 
980222 
960165 

9.960109 
960052 
959995 
959938 
959882 
959825 
959768 
959711 
959654 
959596 
.959539 
959482 
959425 
959368 
959310 
959253 
959195 
959138 
959081 
969023 

9.958965 


958850 
958792 
958734 
968677 
958619 
958561 
958503 
958445 

9.958387 
958329 
958271 
958213 
958154 
958096 
958038 
957979 
967921 
957863 

9.967804 
957746 
957687 
967628 
957570 
957511 
957462 
957393 
J.'57335 
957276 


Sine. 


9.4 

9.4 

9.4 

9.4 

9.4 

9.4 

9.4 

9.4 

9.4 

9.4 

9.4 

9.6 

9,5 

9.5 

9.5 

9.5 

9.5 

9.5 

9.5 

9.5 

9.5 

9.6 

9.5 

9.5 

9.5 

9.6 

9.6 

9.6 

9.6 

9.6 

9.6 

9.6 

9.6 

9.6 

9.6 

9.6 

9,6 

9,6 

9.6 

9,7 

9.7 

9.7 

9.7 

9.7 

9.7 

9.7 

9.7 

9,7 

9.7 

9.7 

9.7 

9.7 

9.8 

9.8 

9.8 

9.8 

9.8 

9.8 

9.8 

9.8 


Tang. 


9,648683 
648923 
649263 
649602 
649942 
650281 
650620 
660959 
661297 
651636 
661974 

9.652312 
652650 
652988 
663326 
653663 
654000 
654337 
654174 
656011 
656348 

9.656684 
666020 
656356 
666692 
657028 
657364 
657699 
658034 
658369 
658704 

9,659039 
659373 
659708 
660042 
660376 
660710 
661043 
661377 
681710 
662043 

9,662376 
662709 
663042 
663375 
663707 
664039 
664371 
664703 
665035 
665366 

9.665697 
666029 
666360 
666691 
667021 
667362 
667682 
668013 
668343 
668672 


D.  10" 


56.6 

66.6 

56.6 

56.6 

66.5 

66.5 

59.5 

56.4 

66.4 

56.4 

56.3 

56.3 

66.3 

56.3 

56.2 

56.2 

56.2 

56.1 

56.1 

56.1 

56.1 

56.0 

56.0 

56.0 

55.9 

55.9 

56.9 

55.9 

55.8 

55.8 

55.8 

56,8 

55.7 

66,7 

55.7 

55.7 

55.6 

56,6 

65,6 

55,5 

55,5 

55.5 

55,4 

55.4 

55.4 

55.4 

55.3 

56.3 

65,3 

55.3 

65.2 

55.2 

55.2 

56,1 

56,1 

55,1 

55.1 

55.0 

55.0 

56.0 


Cotang. 

10.361417 
351077 
350737 
350398 
350058 
349719 
349380 
349041 
348703 
348364 
348026 

10.347688 
347350 
347012 
346674 
346337 
346000 
345663 
346326 
344989 
344662 

10.344316 
343980 
343644 
343308 
342972 
342636 
342301 
341966 
341631 
341296 
,340961 
340S27 
340292 
339968 
339624 


10 


Cotang. 
Degrees. 


338957 
338623 
338290 
337957 

10.337624 
337291 
336968 
336626 
336293 
335961 
336629 
335297 
334965 
334634 

10.334303 
333971 
333620 
333309 
332979 
332648 
332318 
331987 
331657 
331328 


iang. 


iN.  sine.  N.  cos. 


40674  91366 1 60 
4070091343  59 
40727  91331 


40753 


91319  57 


140780  91307 
I  40806  91295 
40833  91283 
40860  91272 
40886  91260 
40913  91248 
40939  91236 
40966  91224 
40992  91212 
41019  91200 
41046  91188 
41072  91176 
41098  91164 
41125  91162 
41151  91140 
41178  91128 
41204  91116 
41231  91104 
41267  91092 
4128491080 
41310  91068 
41337  91066 
41363  91044 
41390  91032 
4141691020 
41443191008 
41469  J90996 
41496  90984 
41622  90972 


41549 
41575 
41602 
41628 
41655 
41681 


90960 
90948 
90936 
90924 
90911 
90899  22 


41707^0887 


41734 
41760 
41787 


90875  20 


90863 
90851 


41813'90839 
41840190826 
41866  j90814 
41892  90802 
41919  90790 


41945 
41972 
41998 
42024 


90778 
90766 
90763 
90741 


4205190729 
4207790717 
42104  90704 
42130  90692 
42156  90680 
42183  90668 
42209  90655 
42235  90643 
42262  |90o31 
N.  cos. In. sine. 


46 


Log.  Sines  and  Tangents.    (25"^)    Natural  Sines. 


TABLE  IL 


0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
69 
60 


Sine. 


D.  lO^'l  Cosine,  |D.  10"   Tang. 


9.625948 
626219 
626490 
626760 
627030 
627300 
627570 
627840 
628109 
628378 
628647 

9.628916 
629185 
629453 
629721 


630257 
630524 
630792 
631059 
631326 

9.631593 
631859 
632125 
632392 
632658 
632923 
633189 
633454 
633719 
633984 
634249 
634514 
634778 
635042 
635306 
636570 
636834 
636097 
636360 
636623 
636886 
637148 
637411 
637673 
637935 
638197 
638458 
638720 
638981 
639242 

9.639503 
639764 
640024 
640284 
640544 
640804 
641064 
641324 
641584 
641842 


Cosine. 


45.1 
45.1 
46.1 
46.0 
46.0 
45.0 
44.9 
44.9 
44.9 
44.8 
44.8 
44.7 
44.7 
44.7 
44.6 
44.6 
44.6 
44.6 
44.5 
44.5 
44.5 
44.4 
44.4 
44.4 
44.3 
44.3 
44.3 
44.2 
44.2 
44.2 
44.1 
44.1 
44.0 
44.0 
44.0 
43.9 
43.9 
43.9 
43.8 
43.8 
43.8 
43.7 
43.7 
43.7 
43.7 
43.6 
43.6 
43.6 
43.5 
43.5 
43.5 
43.4 
43.4 
43.4 
43.3 
43.3 
43.3 
43.2 
43.2 
43.2 


9. 


9. 


.957276 
957217 
957158 
957099 
957040 
956981 
956921 
956862 
956803 
956744 
956684 
.956626 
956566 
956506 
956447 
956387 
956327 
956268 
956208 
966148 
956089 
.956029 
955969 
956909 
955849 
955789 
955729 
955669 
955609 
956548 
966.488 
956428 
966368 
966307 
965247 
955186 
955126 
955065 
955005 
964944 
954883 
964823 
954762 
954701 
964640 
954579 
954618 
964467 
964396 
964335 
954274 
954213 
954152 
964090 
964029 
953968 
953906 
963845 
953783 
953722 
953660 


Sine. 


9.8 
9.8 
9.8 
9.8 
9.8 
9.8 
9.9 
9.9 
9.9 
9.9 
9.9 
9  9 
9  9 
9.9 
9.9 
9.9 
9.9 
9.9 
10.0 
10.0 
10.0 
10.0 
10.0 
10.0 
10.0 
10.0 
10.0 
10.0 
10.0 
10.0 
10.0 
10. 
10. 
10. 
10. 
10. 
10. 
10. 
10. 
10. 
10. 
10. 
10. 
10. 
10. 
10. 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.2 
10.3 


9.668673 
669002 
669332 
669661 
669991 
670320 
670649 
670977 
671306 
671634 
671963 

9.672291 
672619 
672947 
673274 
673602 
673929 
674257 
674584 
674910 
675237 

9.676564 
676890 
676216 
676643 
676869 
677194 
677620 
677846 
678171 
678496 
678821 
679146 
679471 
679795 
680120 
680444 
680768 
681092 
681416 
681740 

9.682063 
682387 
682710 
683033 
683356 
683679 
684001 
684324 
684646 
684968 
685290 
685612 
685934 
686255 
686677 


687219 
687540 
687861 
688182 


Cotang. 


D.  10"  Cotang.  [N  .sine.  N.  cos, 


10.331327  142262 
3309981142288 
330668  42315 
3303391  42341 
3300091  42367 
3296801142394 
329351!  42420 
329023: 1  42446 
328694! I  42473 
3283661142499 
328037:,  425'-'5 

10.3277091  42552 
327381 1 1  42578 
3270531 1  42604 
326726 i  42631 
326398:;  42657 
326071;;  42683 
3257431  42709 
3254161:42736 
325090 1 1  42762 
324763  h  42788 

10.324436  42815 
3241101:42841 
323784 1 '42867 
323457  1142894 
323131 1 J 42920 
322806  i  I  4294(> 
322480  I  42972 
322164 
321829 
321504 

10.321179 
320854 
320529 
3202U5 

319880  43182 
319556 
319232 

318908  ij  43261 
3185841;  43287 
318260  43313 

10.3179371:43340 
3176131143366 
3172901  43392 
316967;  43418 
316644!  43445 
3163211  43471 
315999  1 1  43497 
315676  143523 
315354':  43549 
315032:  43575 

10.3147lu:  43602 
3143881:  43628 
314066  : 43654 
313745 
313423 
313102 
312781 
312460 
312139 
311818 


42999 
43025 
43051 
43077 
43104 
43130 
43156 


43209 
43235 


43680 
43706 
4373J^ 
43759 
43785 
43811 
43837 


Tanf 


90631 
90613 
90606 
90594 
90582 
90569 
90557 
90545 
90532 
90520 
90507 
90495 
90483 
90470 
90458 
90446 
90433 
90421 
90408 
90396 
90383 
90371 
90358 
90346 
90334 
90321 
9030y 
90296 
90284 
90271 
90259 
90246 
90233 
90221 
90208 
90196 
90183 
90171 
90158 
90146 
90133 
90120 
90108 
90095 
90082 
90070 
90057 
90045 
90032 
90019 
90007 
89994 
89981 
89968 
89956 
89943 
89930 
89918 
89906 
89892 
89879 


N.  CO?.  X.t'iDe. 


64  Degrees. 


TAELE  II. 


Log.  Sines  and  Tangents.    (26°)    Natural  Sines. 


47 


Sine. 


.641842 
642101 
642360 
642618 
642877 
643135 
643393 
643650 
643908 
644165 
644423 

.644680 
644936 
645193 
645450 
645706 
645962 
646218 
646474 
646729 
646984 

.647240 
647494 
647749 
648004 
648258 
648512 
648766 
649020 
649274 
649527 

.649781 
650034 
650287 
650539 
650792 
651044 
651297 
651549 
651800 
652052 

.652304 
652555 
652806 
653057 
653308 
653558 
653808 
654059 
654309 
654558 

.654808 
655058 
655307 
655556 
655805 
656054 
656302 
656551 
65G799 
657047 


D.  10' 


43, 
43. 
43, 

43, 
43, 
43, 
43, 
42. 
42. 
42, 
42, 
42, 
42, 
42, 
42. 
42, 
42, 
42, 
42. 
42, 
42, 
42. 
42, 
42. 
42. 
42, 
42. 
42, 
42. 
42. 
42. 
42. 
42. 
42, 
42. 
42. 
42. 
42. 
42. 
41. 
41. 
41. 
41, 
41. 
41. 
41. 
41. 
41. 
41. 
41. 
41. 
41. 
41. 
41. 
41. 
41, 
41. 
41. 
41. 
41. 


Cosine. 


Cosine. 


9.953660 
953599 
953537 
953475 
953413 
953352 
953290 
953228 
953166 
953104 
953042 

9.952980 
952918 
952855 
952793 
952731 
952669 
952606 
952544 
952481 
952419 

9.952356 
952294 
952231 
952168 
952106 
952043 
951980 
951917 
951854 
951791 
951728 
951665 
951602 
951539 
951476 
951412 
951349 
951286 
951222 
951159 
951096 
951032 
950968 
950905 
960841 
950778 
950714 
950650 
950586 
950522 

9.950458 
950394 
950330 
950366 
950202 
950138 
950074 
950010 
949945 
949881 
Sine. 


D.  10'' 


10.3 
10.3 
10.3 
10.3 
10.3 
10.3 
10.3 
10.3 
10.3 
10.3 
10.3 
10.4 
10.4 
10.4 
10.4 
10.4 
10.4 
10,4 
10.4 
10.4 
10.4 
10.4 
10.4 
10.4 
10.5 
10.5 
10.5 
10.5 
10,5 
10.5 
10.5 
10.5 
10.5 
10.5 
10.5 
10.5 
10.5 
10;  6 
10.6 
10.6 
10.6 
10.6 
10.6 
10.6 
10.6 
10.6 
10.6 
10.6 
10.6 
10.6 


9. 


Tang. 


688182 
688502 
688823 
689143 
689433 
689783 
690103 
690423 
690742 
691062 
691381 

1.691700 
692019 
692338 
692656 
692975 
693293 
693612 
693930 
694248 
694566 

1.694883 
695201 
695518 
695836 
696153 
696470 
696787 
697103 
697420 
697736 

1.698053 
698369 
698685 
699001 
699316 
699632 
699947 
700263 
700578 
700893 

1.701208 
701523 
701837 
702152 
702466 
702780 
703095 
703409 
703723 
704036 

(.701350 
704GG3 
704977 
705290 
705603 
705916 
706228 
706541 
706854 
7071G6 

Cotang. 


[D.  lO'l 

53.4 
53.4 
53.4 
53.3 
53.3 
53.3 
53.3 
53.3 
53,2 
53.2 
53.2 
53.1 
53.1 
53,1 
53.1 
53.1 
53.0 
53.0 
53.0 
53.0 
52.9 
52.9 
52.9 
52.9 
52.9 
52.8 
52.8 
52.8 
52.8 
52.7 
52.7 
52.7 
52.7 
62,6 
52.6 
52.6 
62.6 
52.6 
9   K 


52.5 

62.5 
52.4 
62.4 
62.4 
52.4 
62.4 
52.3 
62.3 
52.3 
52.3 
52.2 
52.2 
52.2 
52.2 
52.2 
52.1 
52.1 
52.1 
52.1 
52.1 


Cotang.     ;  N.  sine.  N 


10.311818 
311498 
311177 
310857 
310537 
310217 
309897 
309577 
309258 
308938 
308619 

10.308300 
307981 
307662 
307344 
307025 
306707 
306388 
30S070 
305752 
305434 

10.305117! 
304799 
3044S2 
304164 1 
303847  i 
303530 1 
303213 
302897 
302680 
302264 

10-301947 
301631 
301315 
300999 
300GS4 
300368 
300053 
299737 
299422 
299107 

10-298792 
298477 
298163 
297848 
297534 
297220 
296906 
2S6691 
296277 
295964 

10.295650 
295337 
295023 
294710 
294^97 
294084 
293772 
293469 
29^146 
292834 
Tang. 


43837 
43863 
43889 
43916 
43942 
43968 
43994 
44020 
44046 
44072 
44098 
44124 
44161 


89879 
89867 
89854 
89841 
89828 
89816 
89803 
89790 
89777 
89764 
89752 
89739 
89726 


44177  89713 


44203 
44229 
44265 
44281 
44307 
44333 
44359 
44386 
44411 
44437 
44464 
44490 
44616 
44542 
44568 
44694 
44620 
44646 
44672 
44698 
44724 
44750 
44776 
44802 
4482« 
44854 


89700 
89687 
89674 
89662 
89649 
89636 
89623 
89610 
89597 
89584 
89571 
89558 
89646 
89532 
89519 


89480 
89467 
89454 
89441 
89428 
89416 
89402 
89389 
89376 
4488089363 


44906 
44932 
44958 
44984 
45010 
45036 
46062 
45088 


46140 
45166 
45192 
46218 
45243 
45269 
46296 
45321 
45C47 
45373 
45G99 


89503  31 
89493 ! 30 


89360 
89337 
89324 
89311 
89298 
89285 
89272 
89269 


45114  89246 


89232 
89219 
89206 
89193 
89180 
89167 
89153 
89140 
89127 
89114 
89101 


2v.  COS.  N.sine. 


63  Degrees 


48 


Log.  Sines  and  Tangents.    (27°)    Natural  Sines. 


TABLE  n. 


0 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


Sine. 


D.  10' 


9.657047 
657295 
657642 
657790 
658037 
658284 
658531 
658778 
659025 
659271 
659517 

9.659763 
660009 
660255 
660501 
660746 
660991 
661236 
661481 
661726 
661970 

9.662214 
662459 
662703 
662946 
663190 
663433 
663677 
663920 
664163 
664406 

9.664648 
664891 
665133 
665375 
665617 
665859 
666100 
666342 
666583 
666824 

9.667065 
667305 
667546 
667786 
668027 
668267 
668506 
668746 
668986 
669225 

9.669464 
669703 
669942 
670181 
670419 
670658 
67089G 
671134 
671372 
671609 
CJosine. 


41.3 
41.3 
41.2 
41.2 
41.2 
41.2 
41.1 
41.1 
41.1 
41.0 
41.0 
41.0 
40.9 
40.9 
40.9 
40.9 
40.8 
40.8 
40.8 
40.7 
40.7 
40.7 
40.7 
40.6 
40.6 
40.6 
40.5 
40.5 
40.5 
40.5 
40.4 
40.4 
40.4 
40.3 
40.3 
40.3 
40.2 
40.2 
40.2 
40.2 
40.1 
40.1 
40.1 
40.1 
40.0 
40  0 
40.0 
39.9 
39.9 
39.9 
39.9 
39.8 
39.8 
39.8 
39.7 
39.7 
39.7 
39.7 
39.6 
39.6 


Cosine. 


.949881 
949816 
949752 
949688 
949623 
949558 
949494 
949429 
949364 
949300 
949235 

.949170 
949105 
949040 
948975 
948910 
948845 
948780 
948715 
948650 
948684 

.948519 
948454 


9. 


948323 
948257 
948192 
948126 
948060 
947995 
947929 
947863 
947797 
947731 
947665 
947600 
947533 
947467 
947401 
947335 
947269 
947203 
947 \36 
947070 
947004 
946937 
946871 
946804 
946738 
946671 
946604 
946538 
946471 
946404 
946337 
946270 
946203 
946136 
946069 
946002 
945936 


Sine. 


D.  10' 


10.7 
10.7 
10.7 
10.8 
10.8 
10.8 
10.8 
10.8 
10.8 
10.8 
10.8 
10.8 
10.8 
10.8 
10.8 
10.8 
10.8 
10.9 
10.9 
10.9 
10.9 
10.9 
10.9 
10.9 
10.9 
10.9 
10.9 
10.9 
10.9 
11.0 
11.0 
11.0 
11.0 
11.0 
11.0 
11.0 
11.0 
11.0 
11.0 
11.0 
11.0 
11.0 
11.1 
11.1 
11.1 
11.1 
11.1 
11.1 
11.1 
11.1 
11.1 
11.1 
11.1 
11.1 
11.1 
11.2 
11.2 
11.2 
11.2 
11.2 


Tang. 


9.707166 

707478 

707790 

708102 

708414 

708726 

709037 

709349 

709660 

709971 

710282 
9.710593 

710904 

711215 

711525 

711836 

712146 

712456 

712766 

713076 

713386 

,713696 

714005 

714314 

714624 

714933 

715242 

715551 

715860 

716168 

716477 
9.716785 

717093 

717401,-.  o 

717709  f;-^ 

718017 

718325 

718633 

718940 

719248  9}  •;! 

719556  I ^J -2 
9.719862 :^}-^ 

720169  °f-f 

720476  °J-} 

720783 i^j-; 

721089;°} -J 

721396 j^;-; 
721702  °;-;. 

722009 ;°J-^ 
722316^}-" 
722621  I  ^}-X 
.722927 ! 5  -^ 

723232  ^/.-^ 
723538  °X'^ 
723844 
724149 
724-i64 


D.  lO'l  Cotang.  !  N.  sine.  N.  cos. 


52.0 
52.0 
52.0 
52.0 
51.9 
51.9 
51.9 
51.9 
51.9 
51.8 
51.8 
51.8 
51.8 
51.8 
61.7 
51.7 
51.7 
51.7 
51.6 
51.6 
51.6 
61.6 
51.6 
51.5 
51.5 
51.5 
61.5 
51.4 
51.4 
61.4 
51.4 
51.4 
51.3 


61.3 
61.3 
51.3 
61.2 


60.9 
50.9 
50.9 

724759  1^.-1 
725056  °^-^ 
725369 
725674 


50.8 
50.8 


CJctang. 


10.292834 
292622 
292210 
291898 
291586 
291274 
290963 
290651 
290340 
290029 
289718 

10.289407 
289096 
288786 
288475 
288164 
287854 
287544 
287234 
286924 
286614 

10.286304 
285996 
285686 
285376 
285067 
284758 
284449 
284140 
283832 
283623 

10.283215 
282907 
282699 
282291 
281983 
281676 
281367 
281060 
280762 
280445 

10.280138 
279831 
279524 
279217 
278911 
278604 
278Ji98 
277991 
277685 
277379 

10.277073 
276768 
276462 
276166 
275851 
275546 
275241 
274935 
274631 
274326 


45399 
45426 
' 45451 
i 46477 
I  45503 
1 45529 
j  46554 
! 45580 
1 45606 
! 45632 
45658 
45684 
45710 
45736 
46762 
45787 
45813 
45839 
45865 
i  45891 
j 45917 
146942 
145968 
45994 
I  46020 
46046 
46072 
46097 
46123 
46149 
46175 
46201 
46226 
Ij  46252 
46278 
46304 
46330 
46365 
46381 
46407 
46433 
46458 
46484 
46510 
46536 
46561 
4658'. 
4G613 
46639 
46664 
46690 
46716 
46742 
46767 
46793 
I  46819 
i 46844 
46870 
46896 
46921 
46947 


Tang. 


89101 
89087 
89074 
89061 
89048 
89035 
89021 
89008 
88995 
88981 
88968 
88955 
88942 
88928 
88915 
88902 
88888 
88876 
88862 
88848 
88835 
88822 
88808 
88795 
88782 
88768 
88755 
88741 
88728 
88715 
88701 
88688 
88674 
88661 
88647 
88634 
88620 
88607 
88593 
88580 
88566 
88653 
88539 
88526 
88512 
88499 
88485 
88472 
88468 
88445 
88431 
88417 
88404 
88390 
88377 
88363 
88349 
88336 
88322 
88308 
88295 


N.  COS.  .N.piiic. 


62 


TABLE  II. 


Log.  Sines  and  Tangents.    (28°)    Natural  Sines. 


49 


Sine. 


9.671609 
671847 
672084 
672321 
672568 
672795 
673032 
673268 
673506 
673741 
673977 

9.674213 
674448 
674684 
674919 
675155 
675390 
675624 
675859 
676094 
676328 

9.676562 
676796 
677030 
677264 
677498 
677731 
677964 
678197 
678430 
678663 
678895 
679128 
679360 
679592 
679824 
680056 
680288 
680519 
680750 
680982 

9.681213 
681443 
681674 
681905 
682135 
682365 
682595 
682825 
683055 
683284 

9.683514 
683743 
683972 
684201 
684430 
684658 
684887 
685115 
685343 
685571 


Cosine. 


D.  10" 


39.6 
39.5 
39.5 
39.5 
39.5 
39.4 
39.4 
39.4 
39.4 
39.3 
39.3 
39.3 
39.2 
39.2 
39.2 
39.2 
39.1 
39.1 
39.1 
39.1 
39.0 
39.0 
39.0 
39.0 
38.9 
38.9 
38.9 
38.8 
38.8 
38.8 
38.8 
38.7 
38.7 
38.7 
38.7 
38.6 
38.6 
38.6 
38.6 
38.5 
38.5 
38.5 
38.4 
38.4 
38.4 
38.4 
38.3 
38.3 
38.3 
38.3 
38.2 
38.2 
38.2 
38.2 
38.1 
38.1 
38.1 
38.0 
38.0 
38.0 


Cosine. 


.945935 
945868 
945800 
946733 
945666 
945598 
945631 
945464 
946396 
945328 
945261 

.945193 
945126 
946068 
944990 
944922 
944864 
944786 
944718 
944650 
944582 

.944514 
944446 
944377 
944309 
944241 
944172 
944104 
944036 
943967 


.943830 
943761 
943693 
943624 
943566 
943486 
943417 
943348 
943279 
943210 

.943141 
943072 
943003 
942934 
942864 
942795 
942726 
942656 
942587 
942517 

.942448 
942378 
942308 
942239 
942169 
942099 
942029 
941959 
941889 
941819 


Sine. 


D.  10' 


Tang. 


.725674 
725979 
726284 
726588 
726892 
727197 
727601 
727805 
728109 
728412 
728716 

.729020 
729323 
729626 
729929 
730233 
730635 
730838 
731141 
731444 
731746 

.732048 
732351 
732663 
732955 
733257 
733558 
733860 
734162 
734463 
734764 

.735066 
736367 
735668 
736969 
736269 
736670 
736871 
737171 
737471 
737771 

. 738071 
738371 
738671 
738971 
739271 
739670 
739870 
740169 
740'468 
740767 

.741066 
741365 
741664 
741962 
742261 
742559 
742868 
743156 
743454 
743752 


Cotang. 


D.  10" 


50.8 
60.8 
50  7 
60.7 
60.7 
50.7 
50.7 
50.6 
50.6 
60.6 
50.6 
50.6 
50.6 
60.5 
50.6 
60.5 
50.5 
50.4 
50.4 
50.4 
60.4 
60.4 
50.3 
50.3 
50.3 
60.3 
60.3 


50 

60 

60 

50 

50 

50 

50 

60 

50.1 

60.1 

50.1 

60.0 

50.0 

50.0 

50.0 

60.0 

49.9 

49.9 

49.9 

49.9 

49.9 

49.9 

49.8 

49.8 

49.8 

49.8 

49.8 

49.7 

49.7 

49.7 

49.7 

49.7 

49.7 


Cotang. 


10.274326 
274021 
273716 
273412 
273108 
272803 
272499 
272196 
271891 
271588 
271284 

10.270980 
270677 
270374 
270071 
269767 
269465 
269162 
268859 
268666 
268254 

10.267952 
267649 
267347 
267046 
266743 
266442 
266140 
266838 
265537 
265236 

10.264934 
26^1633 
264332 
264031 
263731 
263430 
263129 
262829 
262529 
262229 

10.261929 
261629 
261329 
261029 
260729 
260430 
260130 
259831 
269532 
259233 

10.258934 
258635 
258336 
258038 
257739 
257441 
257142 
256844 
256546 
256248 


N.  sine.  N.  cos 


46947 
46973 
46999 
47024 
47050 
47076 
47101 
I  47127 


88296 
88281 
88267 
88264 
88240 
88226 
88213 
88199 


47153  88185 


47178 
47204 
47229 
47255 
47281 
47306 
47332 
47358 
47383 


47409  88048 

47434  88034 

47460  88020 

47486  88006 

47511  87993 

47537  87979 

47562  87965 

47588  87951 

47614  87937 

47639  87923 

,  47665  87909 

I  47690  87896 

147716  87882 

!  1 47741  87868 

1 .47767  87854 

j  1 47793  87840 

:  47818  87826 

147844  87812 

1147869  87798 

i;  47895  87784 

^(47920  87770 

I  47946  87766 


88172 
88158 
88144 
88130 
88117 
88103 
88089 
88075 
88062 


47971 
47997 
48022 
48048 
48073 
48099 
48124 
48150 
48176 
48201 
48226 
48252 
48277 
i  48303 
i  48328 


!  1 48354  87632 

;  48379  87518 

1 1 48405  87504 

48430  87490 

48456  87476 

4848187462 


Tan'jc. 


87743 
87729 
87716 
87701 
87687 
87673 
87659 
87645 
87631 
87617 
87603 
87589 
87575 
87561 
87546 


N.  COS.  N.sine. 


61  Degrees. 


50 


Log.  Sines  and  Tangents.    (29°)    Natural  Sines. 


TABLE  n. 


0 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
26 
26 
27 
28 
29 
30 
31 
32 
33 
34 
36 
36 
37 
38 
39 
40 
41 
42 
43 
44 
46 
46 
47 
48 
49 
50 
61 
62 
63 
54 
56 
56 
67 
68 
69 
60 


Sine. 


D.  10" 


9.685571 
685799 
686027 
686254 
686482 
686709 
686936 
687163 
687389 
687616 
687843 

9.688069 
688296 
688621 
688747 
688972 
689198 
689423 
689648 
689873 
690098 
690323 
690548 
690772 
690996 
691220 
691444 
691668 
691892 
692116 
692339 

9.692562 
692785 
693008 
693231 
693453 
693676 
693898 
694120 
694342 
694564 

9.694786 
695007 
696229 
695450 
696671 
695892 
696113 
696334 
696554 
696776 

9.696995 
697216 
697436 
697654 
697874 
698094 
698313 
698532 
698761 
698970 


Cosine. 


38,0 
37.9 
37.9 
37.9 
37.9 
37.8 
37.8 
37.8 
37.8 
37.7 
37.7 
37.7 
37.7 
37.6 
37.6 
37.6 
37.6 
37.6 
37.5 
37.5 
37.6 
37.4 
37.4 
37.4 
37.4 
37.3 
37.3 
37.3 
37.3 
37.5 
37.2 
37.2 
37.1 
37.1 
37.1 
37.1 
37.0 
37.0 
37.0 
37.0 
36.9 
36.9 
36.9 
36.9 
36.8 
36.8 
36.8 
36.8 
36.7 
36.7 
36.7 
36.7 
36.6 
36.6 
36.6 
36.6 
36.6 
36.5 
36.6 
36.6 


Cosine.  {D.  10^' 


,941819 
941749 
941679 
941609 
941639 
941469 
941398 
941328 
941258 
941187 
941117 
,941046 
940975 
940905 
940834 
940763 
940693 
940622 
940661 
940480 
940409 
.940338 
940267 
940196 
940125 
940054 
939982 
939911 
939840 
939768 
939697 
,939626 
939664 
939482 
939410 
939339 
939267 
939196 
939123 
939052 
938980 
.938908 
938836 
938763 
938691 
938619 
938547 
938476 
938402 
938330 
938268 
.938185 
938113 
938040 
937967 
937895 
937822 
937749 
937676 
937604 
937631 


Sine. 


11.7 

11.7 

11.7 

11.7 

11 

11 

11 

11 

11 

11.7 

11.7 

11.8 

11.8 

11.8 

11.8 

11.8 

11.8 

11.8 

11.8 

11.8 

11.8 

11.8 

11.8 

11.8 

11.9 

11.9 

11.9 

11.9 

11.9 

11.9 

11.9 

11.9 

11.9 

11.9 

11.9 

11.9 

12.0 

12,0 

12.0 

12.0 

12.0 

12.0 

12.0 

12.0 

12.0 

12.0 

12.0 

12.0 

12.1 

12.1 

12.1 

12.1 

12.1 

12.1 

12.1 

12.1 

12.1 

12.1 

12.1 

12.1 


Tang. 


.743762 
744060 
744348 
744646 
744943 
745240 
745638 
745835 
746132 
746429 
746726 

. 747023 
747319 
747616 
747913 
748209 
748506 
748801 
749097 
749393 


9.7i 


749986 
760281 
760676 
750872 
761167 
751462 
751767 
762062 
762347 
762642 
752937 
763231 
763626 
753820 
764116 
754409 
754703 
764997 
765291 
755585 
755878 
766172 
756466 
756769 
767062 
767346 
767638 
757931 
758224 
768517 
758810 
759102 
769396 
769687 
769979 
760272 
760564 
760856 
761148 
761439 


Cotang. 


D.  10" 


49.6 

49.6 

49.6 

49.6 

49.6 

49.6 

49.6 

49.5 

49.5 

49.5 

49.6 

49.4 

49.4 

49.4 

49.4 

49.4 

49.3 

49.3 

49.3 

49.3 

49.3 

49.3 

49,2 

49.2 

49.2 

49.2 

49.2 

49.2 

49.1 

49.1 

49.1 

49.1 

49.1 

49.1 

49,0 

49.0 

49,0 

49.0 

49.0 

49.0 

48.9 

48.9 

48,9 

48.9 

48.9 

48.9 

48,8 

48.8 

48.8 

48.8 

48.8 

48. 8i 

48.7  1 

48.7 

48.7 

48.7 

48.7 

48.7 

48.6 

48.6 


Cotang.     I  ;N.  sine.  N.  cos 


10. 


10 


10 


10 


10 


10 


256248 
255950 
255652 
255355 
265067 
264760 
254462 
254165 
253868 
253571 
263274 
262977 
252681 
252384 
252087 
251791 
251495 
261199 
260903 
250607 
250311 
250015 
249719 
249424 
249128 
248833 
248538 
248243 
247948 
247653 
247358 
247063 
246769 
246474 
246180 
245885 
245591 
246297 
246003 
244709 
244415 
244122 
243828 
243636 
243241 
242948 
242655 
242362 
242069 
241776 
241483 
,241190 
240898 
240605 
240313 
240021 
239728 
239436 
239144 
2S8852 
238561 


'48481 
148506 
: 48532 
'[  48557 

48583 
: 48608 

48634 


87462 
87448 
87434 
87420 
87406 
87391 
87377 


4865987363 
48684|87349 
48710j87335 
4873587321 
48761187306 
48786187292 


48811 
48837 
48862 
48888 
48913 
48938 
48964 


87278 
87264 
87250 
87235 
87221 
87207 
87193 


48989187178 
4901487164 
49040!87150 


149065 
; 49090 
1 49116 
49141 


87136 
87121 
87107 
87093 


49166  87079 


49192 
49217 
49242 
49268 
49293 


87064 
87050 
87036 
87021 
87007 


49318  86993 

4934486978 

49369  86964 

4939486949 

4941986935 

49445186921 

49470!86906 

!  49495186892 

'  49.521 186878 

49546  86863 

i;  49571  86849 

1 1 49596186834 

1 1 49622186820 

ij  49647  J86805 

i49672!86791 

|49697j86777 

i  49723  86762 

:  49748 186748 

''49773  86733 

49798'86719 

;  4982486704 

i  1 49849  86690 

j  149874  86675 

'^  49899,86661 

49924^86646 

|i  49950  86632 

i  49975186617 

i  50000186603 


I   Tang.   ||N.  co«.|]\.piue. 


60  Degrees. 


TABLE  II. 


Log.  Sines  and  Tangents.    (30°)    Natural  Sines. 


51 


Sine. 

9.698970 
699189 


D.  10" 


699407 

699626 

699844 :  ^,, 

700062 i^^ 

700280 

700498 

700716 

700933 

701151 
.701368 

701585 

701802 

702019 

702236 

702452 

702669 

702885 

703101 

703317 
.703533 

703749 

703964 

704179 

704395 

704610 

704825 

705040 

705254 

705469 
.705683 

705898 

706112 

706326 

706539 

706753 

706967 

707180 

707393 

707606 
.707819 

708032 

708245 

708458 

708670 

708882 

709094 

709303 

709518 

709730 
.709941 

710153 

710364 

710575 

710786 

710967 

711208 

711419 

711629 

711839 


136 


Cosine. 


Cosine. 


9.937631 
937458 
937385 
937312 
937238 
937165 
937092 
937019 
936946 
936872 
936799 
936725 
936652 
936578 
936505 
936431 
936357 
936284 
936210 
936136 
936062 
935988 
935914 
935840 
935766 
935692 
935618 
935543 
935469 
935395 
935320 

9.935246 
935171 
935097 
935022 
934948 
934873 
934798 
934723 
934649 
934574 
934499 
934424 
934349 
934274 
934199 
934123 
934048 
933973 
933898 
933822 

9.933747 
933671 
933596 
933520 
933445 
933369 
933293 
933217 
933141 
933066 


Sine. 


D.  10" 


12.1 

12.2 

12.2 

12.2 

12.2 

12.2 

12.2 

12 

12 

12 

12 

12 

12 

12 

12 

12.3 

12.3 

12.3 

12.3 

12.3 

12.3 

12.3 

12.3 

12.3 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.4 

12.5 

12.5 

12.5 

12.5 

12.5 

12.5 

12.5 

12.5 

12.5 

12.5 

12.5 

12.5 

12.6 

12.6 

12.6 

12.6 

12.6 

12.6 

12.6 

12.6 

12.6 

12.6 

12.6 


9. 


Tang. 


9. 


9. 


761439 
761731 
762023 
762314 
762605 
762897 
763188 
763479 
763770 
764061 
764352 
764643 
764933 
765224 
765514 
765805 
766095 
766385 
766675 
766965 
767255 
767545 
767834 
768124 
768413 
768703 
768992 
769281 
769570 
769860 
770148 
770437 
770726 
771015 
771303 
771592 
771880 
772168 
772457 
772745 
773033 
773321 
773608 
773896 
774184 
774471 
774759 
775046 
775333 
775621 
775908 
776195 
776482 
776769 
777055 
777342 
777628 
777915 
778201 
778487 
778774 


Cotang. 


D.  10' 


48.6 
48.6 
48.6 
48.6 
48.5 
48.5 
48.5 
48.5 
48.5 
48.5 
48.4 
48.4 
48.4 
48.4 
48.4 
48.4 
48.4 
48.3 
48.3 
48.3 
48.3 
48.3 
48.3 
48.2 
48.2 
48.2 
48.2 
48.2 
48.2 
48.1 
48.1 
48.1 
48.1 
48.1 
48.1 
48.1 
48.0 
48.0 
48.0 
48.0 
48.0 
48.0 
47.9 
47.9 
47.9 
47.9 
47.9 
47.9 
47.9 
47.8 
47.8 
47.8 
47.8 
47.8 
47.8 
47.8 
47,7 
47.7 
47.7 
47.7 


Cotang.     |,N.  sine.  N.  cos 


86603 


10.238561! 
238269 : 
237977 ' 
237686 ! 
237394 ' 
237103  I 
236812  I 
236521  I 
236230  I 
235939  I 
235648  I 

10.235357 
235087 1 
234776  I 
234486  i 
234195 ! 
233905 
233615 
233325 
233035 
232745 

10.232455 
232166 
231876 
231587 
231297 
231008 
230719 
230430 
230140 
229852 

10.229563 
229274 
228985 
228697 
228408 
228120 
227832 
227543 
227255 
226967 

10-226679 
226392 
226104 
225816 
225529 
225241 
224954 
224667 
224379 
224092 

10.223805 
223518 
223231 
222945 
222658 
222372 
222085 
221799 
221612 
221226 


Tang. 


50000 

50025 

50050 

5007{) 

50101 

50126 

50151 

50176 

5a201 

60227 

50252 

5027 

50302 

50327 

50362 

60377 

60403 

50428 

60463 

50478 

50503 

50528 

50563 

50578 

50603 

60628 

50664 

50679 

50704 

50729 

50754 

50779 

50804 

50829 

50854 

50879 

50904 

50929 

50954 

60979 

51004 

61029 

51054 

61079 

61104 

51129 

51154 

51179 

51204 

61229 

51254 

51279 

61304 

61329 

51354 

51379 

51404 

51429 

51454 

51479 

51504 


86573 
86559 
86544 
86530 
86515 
86601 
86486 
80471 
86457 
86442 
86427 
86413 
86398 
86384 
86369 
86354 
86340 
86326 
86310 
86295 
86281 
86266 
86251 
86237 
86222 
86207 
86192 
86178 
86163 
86148 
86133 
86119 
86104 
86089 
86074 
86059 
86045 
86030 
86015 
86000 
85985 
85970 
85956 
85941 
85926 
85911 
85896 
85881 
85866 
85861 
85836 
85821 
85806 
85792 
86777 
85762 
85747 
85732 
85717 
N.  cos.  N.sine. 


59  Degrees. 


21 


52 


Log.  Sines  and  Tangents.    (31°)    Natural  Sines. 


TABLE  II. 


Sine.  |D.  10' 


0 
1 
2 
3 
4 

0 

6 

7 

8 

9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30  j 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


.711839 
712050 
712260 
712469 
712679 
712889 
713098 
713308 
713517 
713726 
713935 

.714144 
714352 
714561 
714769 
714978 
715186 
715394 
715602 
715809 
716017 

.716224 
716432 
716639 
716846 
717053 
717259 
717466 
717673 
717879 
718085 

.718291 
718497 
718703 
718909 
719114 
719320 
719525 
719730 
719935 
720140 

.720345 
720549 
720754 
720958 
721162 
721366 
721670 
721774 
721978 
722181 

.722385 
722588 
722791 
722994 
723197 
723400 
723603 
723805 
724007 
724210 

Cosine. 


35.0 
33.0 
35.0 
34.9 
34.9 
34.9 
34.9 
34.9 
34.8 
34.8 
34.8 
34.8 
34.7 
34.7 
34.7 
34.7 
34.7 
34.6 
34.6 
34.6 
34.6 
34.5 
34.5 
34.5 
34.5 
34.5 
34.4 
34.4 
34.4 
34.4 
34.3 
34.3 
34.3 
34.3 
34.3 
34.2 
34.2 
34.2 
34.2 
34.1 
34.1 
34.1 
34.1 
34.0 
34.0 
34  0 
34.0 
34.0 
33.9 
33.9 
33.9 
33.9 
33.9 
33.8 
33.8 
33.8 
33.8 
33.7 
33.7 
33.7 


Cosine. 

.933086 
932990 
93-2914 
932833 
932762 
932685 
932609 
932533 
932457 
932380 
932304 

.932228 
932151 
932075 
931998 
931921 
931845 
931768 
931691 
931614 
931537 

.931460 
931383 
931306 
931229 
931152 
931076 
930998 
930921 
930843 
930766 

.930688 
930611 
930533 
930456 
930378 
930300 
930223 
930145 
930067 
929989 

1.929911 
929833 
929755 
929677 
929599 
929521 
929442 
929364 
929286 
929207 

1.929129 
929050 
928972 
928893 
928815 
928736 
928667 
928678 
928499 
928420 


Sine. 


D.  10"   Tang. 


12.6 
12.7 
12.7 
12,7 
12.7 
12.7 
12.7 
12.7 
12.7 
12.7 
12.7 
12.7 
12.7 
12.8 
12.8 
12.8 
12.8 
12,8 
12.8 
12.8 
12.8 
12.8 
12.8 
12.8 
12.9 
12.9 
12.9 
12.9 
12.9 
12.9 
12.9 
12.9 
12.9 
12.9 
12.9 
12.9 
13.0 
13.0 
13.0 
13.0 
13.0 
13.0 
13.0 
13.0 
13.0 
13.0 
13.0 
13.0 
13.1 
13.1 
13.1 
13.1 
13.1 
13.1 
13.1 
13.1 
13.1 
13.1 
13.1 
13.1 


.778774 
779060 
779346 
779632 
779918 
780203 
780489 
780776 
781060 
781346 
781631 

.781916 
782201 
782486 
782771 
783056 
783341 
783626 
783910 
784195 
784479 

.784764 
785048 
785332 
785616 
785900 
786184 
786468 
786752 
787036 
787319 

.787603 
787886 
788170 
788453 
788736 
789019 
789302 
789585 
789868 
790151 

.790433 
790716 
790999 
791281 
791563 
791846 
792128 
792410 
792692 
792974 

i.  793256 
793538 
793819 
794101 
794383 
794664 
794945 
795227 
795508 
795789 


Cotang. 


D.  10" 


47.7 
47.7 
47.6 
47.6 
47.6 
47.6 
47.6 
47.6 
47.6 
47.5 
47.5 
47.5 
47.5 
47.5 
47.5 
47.5 
47.5 
47.4 
47.4 
47.4 
47.4 
47.4 
47.4 
47.3 
47.3 
47.3 
47.3 
47.3 
47.3 
47.3 
47.2 
47.2 
47.2 
47.2 
47.2 
47.2 
47.2 
47.1 
47.1 
47.1 
47.1 
47.1 
47.1 
47.1 
47.1 
47.0 
47.0 
47.0 
47.0 
47.0 
47.0 
47.0 
46.9 
46.9 
46.9 
46.9 
46.9 
46.9 
46.9 
46.8 


Cotang. 


10.221226 
220940 
220654 
220368 
220082 
219797 
219511 
219225 
218940 
218654 
218369 

10.218084 
217799 
217514 
217229 
216944 
216639 
216374 
216090 
215805 
215621 

10.216236 
214952 
214668 
214384 
214100 
213816 
213532 
213248 
212964 
212681 

10.212397 
212114 
211830 
211647 
211264 
210981 
210698 
210415 
210132 
209849 

10.209667 
209284 
209001 
208719 
208437 
208164 
207872 
207590 
207308 
207026 

10.206744 
206462 
206181 
205899 
205617 
205336 
205055 
204773 
204492 
204211 


N.sine.jA.  cos.i 

51504185717 
51529!o5702 
51o54!d5687 
515/9|»5672 
51604185657 
51628185642 
51653  65627 
51678  85612 
51703  83397 
51728  83382 
51733  83367 
51778185551 
51803 185536 
51828I85521 
51852186506 
51877185491 
61'902j83476 
51927185461 
51952I85446 
51977185431 
52002183416 
62026185401 
52051 185385 
52076|85370 
52101  85355 
152126  86340 
152151185325 
!52175|85310 
i  52200185294 


1 52225 
52230 
1 52276 
1 52299 
1 52324 
1 52349 
J  52374 
1 52399 
1 52423 
62448 
52473 
52498 


85279 
85264 
85249 
86234 
85218 
85203 
85188 
85173 
86157 
85142 
85127 
85112 


52522  85096 


52547 

62672 
52697 
52621 
52646 


85081 
83066 
85U31 
83036 
85U20 


52671185005 

\  5269b!84989 
84974 
84939 
84943 
84928 
84913 
84897 
84882 


i52i20 
1 52746 
152770 
1 52794 
! 62819 
i  52844 
ii528o9 


52893184866 


Tang. 


1152918 
1152943 
1152967 

1 1 52992 


84851 
84836 
84820 
84806 


N.siae. 


58  Degrees. 


TABLE  n. 


Log.  Sines  and  Tangents.    (32°)    Natural  Sines. 


53 


Sine. 

9.724210 
724412 
724614 
72.i816 
725017 
725219 
725420 
725922 
725823 
726024 
726225 

9.726426 
726626 
726827 
727027 
727228 
727428 
727628 
727828 
728027 
728227 

9.728427 
728626 
728825 
729024 
729223 
729422 
729621 
729820 
730018 
730216 
730415 
730613 
730811 
731009 
731206 
731404 
731602 
731799 
731996 
732193 

9.732390 
732587 
732784 
732980 
733177 
733373 
733569 
733765 
733961 
734157 

9.734353 
734549 
734744 
734939 
735135 
735330 
736526 
735719 
735914 
736109 


Cosine. 


D^10"|    Cosine.    |D.  10' 


33.7 
33.7 
33.6 
33.6 
33.6 
33.6 
33.6 
33.6 
33.6 
33.6 
33.5 
33.4 
33.4 
33.4 
33.4 
33.4 
33.3 
33.3 
33.3 
33.3 
33.3 
33.2 
33.2 
33.2 
33.2 
33.1 
33.1 
33.1 
33.1 
33.0 
33.0 
33.0 
33.0 
33.0 
32.9 
32.9 
32.9 
32.9 
32.9 
32.8 
32.8 
32.8 
32.8 
32.8 
32.7 
32.7 
32.7 
32.7 
32.7 
32.6 
32.6 
32.6 
32.6 
32.6 
32.6 
32.6 
32.6 
32,6 
32.4 
32.4 


.928420 
928342 
928263 
928183 
928104 
928025 
927946 
927867 
927787 
927708 
927629 

.927549 
927470 
927390 
927310 
927231 
927151 
927071 
926991 
92691 1 
926831 

.926751 
926671 
926591 
926511 
926431 
926351 
926270 
926190 
926110 
926029 

.925949 
925868 
925788 
925707 
925626 
925545 
926466 
925384 
925303 
925222 

1.926141 
925060 
924979 
924897 
924816 
924735 
924654 
924572 
924491 
924409 

1.924328 
924246 
924164 
924083 
924001 
923919 
923837 
923755 
923673 
923591 


Sine. 


13.2 
13.2 
13.2 
13.2 
13.2 
13.2 
13.2 
13.2 
13.2 
13.2 
13.2 
13.2 
13.3 
13.3 
13.3 
13.3 
13.3 
13.3 
13.3 
13.3 
13.3 
13.3 
13.3 
13.3 
13.4 
13.4 
13.4 
13.4 
13.4 
13.4 
13.4 
13.4 
13.4 
13.4 
13.4 
13.4 
13.6 
13.6 
13.5 
13.5 
13.5 
13.6 
13.5 
13.5 
13.5 
13.6 
13.6 
13.6 
13.6 
13.6 
13.6 
13.6 
13.6 
13.6 
13.6 
13.6 
13.6 
13.6 
13.7 
13.7 


Tang. 


D.  10"!  Cotang.  |  N.  sine.  N.  cos 


,796789 
796070 
796351 
796632 
796913 
797194 
797475 
797756 
798036 
798316 
798596 

.798877 
799157 
799437 
799717 
799997 
800277 
800567 
800836 
801116 
801396 

.801676 
801966 
802234 
802513 
802792 
803072 
803361 
803630 
803908 
804187 

.804466 
804746 
805023 
805302 
805580 
805859 
806137 
806416 
806693 
806971 

.807249 
807527 
807805 
808083 
808361 
808638 
808916 
809193 
809471 
809748 

.810026 
810302 
810580 
810857 
811134 
811410 
811687 
811964 
812241 
812517 


Cotang. 


46.8 
46.8 
46.8 
46.8 
46.8 
46.8 
46.8 
46.8 
46.7 
46.7 
46.7 
46.7 
46.7 
46.7 
46.7 
46.6 
46.6 
46.6 
46.6 
46.6 
46.6 
46.6 
46.6 
46.5 
46.5 
46.5 
46.5 
46.6 
46.6 
46.6 
46.6 
46.4 
46,4 
46.4 
46.4 
46.4 
46,4 
46.4 
46.3 
46.3 
46.3 
46.3 
46.3 
46.3 
46.3 
46.3 
46.2 
46,2 
46.2 
46.2 
46.2 
46.2 
46.2 
46.2 
46.2 
46.1 
46,1 
46.1 
46.1 
46.1 


10.204211  I 
203930 ! 
203649  i 
203368 
203087  I 
202806 
202625 
202245 
201964 
201684 
201404 

10.201123 
200843 
200563 
200283 
200003 
199723 
199443 
199164 
198884 
198604 

10.198326 
198045 
197766 
197487 
197208 
196928 
196649 
196370 
196092 
195813 

10.195534 
195256 
194977 
194698 
194420 
194141 
193863 
193685 
193307 
193029 

10.192761 
192473 
192195 
191917 
191639 
191362 
191084 
190807 
190529 
190252 

10.189975 
189698 
189420 
189143 
188866 
188590 
188313 
188036 
187759 
187483 


52992 
63017 
63041 
53066 
53091 
53115 


84805 
84789 
84774 
84759 
84743 
84728 


63140J84712 


53164 

53189 
63214 
53238 
53263 
53288 
63312 
53337 


84697 
84681 
84666 
84650 
84635 
84619 
84604 
84588 


5336184573 
5338684557 


53411 
63435 
53460 
53484 
53609 
53534 


84542 
84526 
84511 
84496 
84480 
84464 
53658184448 
63583  84433 
63607j844l7 
5363284402 
53656184386 
6368184370 
53705184365 
53730|84339 
63754!84324 
i  63779184308 
j  53804!84292 
53828;84277 

I  53853184261 
53877^4245 
5390284230 
5392684214 
5395184198 
53975184182 
5400084167 
54G24|84161 
54049|84135 
54073184120 

I I  64097184104 
j  154122:84088 
l!54146j84072 
il64171'840o7 
1 154 195 184041 

I  j  54220  84025 
,  54244  84009 
I  54269 '83994 

54293  83978 
'54317:83962 

54342,83946 
'  64366  83930 
154391  83915 

64415  83899 
i  54440:83883 
1 54464183867 


Tang. 


N.  COS.  N. sine. 


57  Degrees. 


54 


Log.  Sines  and  Tangents.  (33°)  Natural  Sines.     TABLE  II. 


4 

6 

6 

7 

8 

9 

10 

11 

12 

13 

14 

16 

16 

17 

18 

19 

20 

21 

22 

23 

24 

26 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

60 

61 

62 

63 

64 

55 

66 

67 

68 

59 

60 


Sine. 


9.736109 
736303 
736498 
736692 
736886 
737080 
737274 
737467 
737661 
737856 
738048 
9.738241 
738434 
738627 
738820 
739013 
739206 
739398 
739590 
739783 
739975 
740167 
740359 
740550 
740742 
740934 
741126 
741316 
741508 
741699 
741889 
9.742080 
742271 
742462 
742662 
742842 
743033 
743223 
743413 
743602 
743792 
9.743982 
744171 
744361 
744550 
744739 
744928 
745117 
745306 
745494 
745683 
9.745871 
746059 
746248 
746436 
746624 
746812 
746999 
747187 
747374 
747562 


D.  10' 


Cosine. 


32.4 

32.4 

32.4 

32.3 

32.3 

32.3 

32,3 

32.3 

32.2 

32.2 

32.2 

32.2 

32,2 

32.1 

32.1 

32.1 

32.1 

32.1 

32.0 

32.0 

32.0 

32.0 

32.0 

31.9 

31.9 

31.9 

31.9 

31.9 

31.8 

31.8 

31.8 

31.8 

31.8 

31.7 

31.7 

31.7 

31.7 

31.7 

31.6 

31.6 

31.6 

31.6 

31.6 

31.6 

31.6 

31.5 

31.5 

31.6 

31.4 

31.4 

31.4 

31.4 

31.4 

31.3 

31.3 

31.3 

31.3 

31.3 

31.2 

31.2 


Cosine. 


9.923591 
923509 
923427 
923345 
923263 
923181 
923098 
923016 
922933 
922861 
922768 
9.922686 
922603 
922520 
922438 
922356 
922272 
922189 
922106 
922023 
921940 
9.921867 
921774 
921691 
921607 
921624 
921441 
921357 
921274 
921190 
921107 
.921023 
920939 
920856 
920772 
920688 
920604 
920520 
920436 
920352 
920268 
,920184 
920099 
920015 
919931 
919846 
919762 
919677 
919593 
919508 
919424 
919339 
919254 
919169 
919085 
919000 
918916 
918830 
918745 
918669 
918574 


D.  10" 


Sine. 


13.7 

13.7 

13.7 

13.7 

13.7 

13.7 

13.7 

13.7 

13.7 

13.7 

13.8 

13.8 

13.8 

13.8 

13.8 

13.8 

13.8 

13.8 

13.8 

13.8 

13.8 

13.9 

13.9 

13.9 

13.9 

13.9 

13.9 

13.9 

13.9 

13.9 

13.9 

13.9 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.2 

14.2 

14.2 

14.2 


Tang. 


9.812517 
812794 
813070 
813347 
813623 
813899 
814175 
814452 
814728 
815004 
815279 
9.816555 
816831 
816107 
816382 
816668 
816933 
817209 
817484 
817769 
818035 
9.818310 
818686 
818860 
819136 
819410 1 
819684 
819959 
820234 
820508 
820783 
9.821057 
821332 
821606 
821880 
822164 
822429 
822703 
822977 
823250 
823524 
9.823798 
824072 
824345 
824619 
824893 
825166 
825439 
826713 
825986 
826259 
826532 
826805 
827078 
827351 
827624 
827897 
828170 
828442 
828716 
828987  I 


D.  10' 


46.1 
46.1 
46  1 
46.0 
46.0 
46.0 
46.0 
46.0 
46.0 
46.0 
46.0 
45.9 
45.9 
46.9 
45.9 
46.9 
46.9 
45.9 
46.9 
46.9 
45.8 
45.8 
45.8 
|46.8 
45.8 
I45.8 
45.8 
45.8 
45.8 
45.7 
45.7 
45.7 
45.7 
45.7 
45.7 
45.7 
45.7 
46.7 
45.6 
46.6 
46.6 
46.6 
46.6 
45.6 
45.6 
45.6 
45.6 
46.5 
45.6 
46.6 
45.5 
45.6 
46.5 
45.5 
45.5 
45.6 
46.4 
45.4 
45.4 
46.4 


Cotano 


N.  sine 


Cotang. 


10.187482 
187206 
186930 
186653 
186377 
186101 
185825 
185548 
185272 
184996 
184721 

10.184445 
184169 
183893 
183618 
183342 
1830G7 
182791 
182516 
182241 
181966 

10.181690 
181415 
181140 
180866 
180590 
180316 
180041 
179766 
179492 
179217 
178943 
178668 
178394 
178120 
177846 
177571 
177297 
177023 
176750 
176476 
10.176202 
176928 
176655 
176381 
175107 


64464 
54488 
64513 
64537 


83867 
83851 
83835 
83819 


54561183804 
54586183788 
54610j83772 
54635  83756 
54659  fi3740 


54683 
64708 


83724 
83708 


10. 


64732  83692 
54756 183676 
6478183660 
64805  83645 
5482983629 
54854183613 
54878^83597 
54902:83581 
64927  83565 
5495183549 
I  54975 '83533 
54999  ;835 17 
55024:83501 
55048:83485 
!  55072  83469 
:  56097  83453 
155121183437 
55145183421 
56169|83405 
55194I83389 
55218:83373 
55-242  83356 
55266 '83340 
I  65291  83324 
5531683308 
55339  83292 
55363 '83276 
55388  83260 
55412|83244 
55436 '83228 
65460;83212 
55484183196 
55509'83179 
5553383163 
55557:83147 


174834  '5658r83131 
174661  1 55605 '83 115 
174287  ii  55630  83098 
174014 II  55b5  183082 
173741  1 1 1>6678 '83066 
10. 173468  1155702  83060 
173196  11  55726'83034 
172922 1  ;55750'83U17 
172649  :  55775  83001 
172376  j  55799  82985 
172103!  1 55823  82969 
171830!  I  55847  82963 
171658  !j  65871 '82936 
171285  |55&95;82920 
171013  |65919[82904 
X.  cos.lN.sine.l 


Tang. 


60 
59 
58 
57 
66 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 

28 

27 

26 

26 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 


56  Degrees. 


TABLE  II. 


Log.  Sines  and  Tangents.  (34°)  Natural  Sines. 


55 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 


Sine. 


9.747562 
747749 
747936 
748123 
748310 
748497 
748683 
748870 
749056 
749243 
749426 

9.749615 
749801 
749987 
750172 
750358 
750543 
750729 
750914 
751099 
751284 

9.751469 
751654 
751839 
752023 
752208 
752392 
752576 
752760 
752944 
753128 

9.753312 
753495 
753679 
753862  i 
754046  i 
754229 
754412  I 
754595  i 
754778 1 
754960  i 

9.755143! 
755326 
755508 
755690 
755872 
756054 
756236 
756418 
756600 
756782 
9.756963 
757144 
757326 
757507 
757688 
757869 
758050 
758230 
758411 , 

J758591_j 
Cosine.  I 


D.  10"|  Cosine. 


31.2 

31.2 

31.2 

31.1 

31.1 

31.1 

31.1 

31.1 

31.0 

31.0 

31.0 

31.0 

31.0 

30.9 

30.9 

30.9 

30.9 

30.9 

30.8 

30.8 

30.8 

30.8 

30.8 

30.8 

30.7 

30.7 

30.7 

30.7 

30.7 

30.6 

30.6 

30.6 

30.6 

30.6 

30.5 

30.5 

30.5 

30  5 

30.5 

30.4 

30.4 

30.4 

30.4 

30.4 

30.4 

30.3 

30.3 

30.3 

30.3 

30.3 

30.2 

30.2 

30.2 

30.2 

30.2 

30.1 

30.1 

30.1 

30.1 

30.1 


918574 
918489 
918404 
918318 
918233 
918147 
918062 
917976 
917891 
917805 
917719 
9.917634 
917548 
917462 
917376 
917290 
917204 
917118 
917032 
916946 
916859 
916773 
916687 
916600 
916514 
916427 
916341 
916254 
916167 
916081 
915994 
9.915907 
915820 
916733 
915646 
915559 
915472 
915385 
915297 
915210 
915123 
915035 
914948 
914860 
914773 
914685 
914598 
914510 
914422 
914334 
914246 
9.914158 
914070 
913982 
913894 
913806 
913718 
913630 
913541 
913453 
913365 
Sine. 


D.  10" 


14.2 

14.2 

14.2 

14.2 

14.2 

14.2 

14,2 

14.3 

14.3 

14.3 

14.3 

14.3 

14.3 

14.3 

14,3 

14.3 

14.3 

14.4 

14.4 

14.4 

14.4 

14.4 

14.4 

14.4 

14.4 

14.4 

14.4 

14,4 

14.5 

14.5 

14.5 

14.5 

14.5 

14.5 

14.5 

14.5 

14,5 

14.5 

14.5 

14.5 

14.6 

14,6 

14.6 

14.6 

14.6 

14.6 

14.6 

14.6 

14.6 

14.6 

14.7 

14.7 

14.7 

14,7 

14.7 

14.7 

14.7 

14,7 

14.7 

14.7 


Tang. 


9.828987 
829260 
829532 
829805 
830077 
830349 
830621 
830893 
831165 
831437 
831709 

9.831981 
832253 
832525 
832796 
833068 
833339 
833611 
833882 
834154 
834425 
,834696 
834967 
835238 
835509 
835780 
836051 
836322 
836593 
836864 
837134 
837405 
837675 
837946 
838216 
838487 
838757 
839027 
839297 
839568 
839838 
9.840108 
840378 
840647 
840917 
841187 
841457 
841726 
841996 
842266 
842535 
842805 
843074 
843343 
843612 
843882 
844151 
844420 
844G89 
844958 
846227 


D.  10"  Cotang.  1  N.sine.  N.  cos 


Cotang. 


45.4 

45.4 

45.4 

46.4 

45.4 

45.3 

45.3 

45.3 

45.3 

45.3 

45.3 

45.3 

45.3 

45.3 

45.3 

45.2 

45.2 

45.2 

45.2 

45.2 

45.2 

45.2 

45.2 

45.2 

45.2 

45.1 

45,1 

45.1 

45.1 

45.1 

45,1 

45.1 

45.1 

45.1 

45.1 

45.0 

45.0 

45.0 

45.0 

45.0 

45.0 

45.0 

45.0 

45.0 

44.9 

44.9 

44.9 

44.9 

44.9 

44.9 

44.9 

44.9 

44.9 

44.9 

44.9 

44.8 

44.8 

44.8 

44.8 

44.8 


10.171013 
170740 
170468 
170195 
169923 
169651 
169379 
169107 
168835 
168563 
168291 

10.168019 
167747 
167475 
167204 
166932 
166661 


166118 
165846 
165575 

10.165304 
165033 
164762 
164491 
164220 
163949 
163678 
163407 
163136 
162866 

10.162595 
162325 
162054 
161784 
161513 
161243 
160973 
160703 
160432 
160162 

10.159892 
159622 
159353 
159083 
158813 
158643 
168274 
158004 
157734 
157465 

10.157195 
166926 
166667 
156388 
156118 
165849 
155580 
165311 
156042 
154773 


Tang. 


55919 

55943 

55968 

55992 

56016 

56040 

56064 

56088 

56112 

56136 

56160 

56184 

56208 

56232 

56256 

56280 

56305 

56329 

66363 

56377 

56401 

56425 

56449 

56473 

56497 

56621 

66545 

56669 

56593 

56617 

56641 

66666 

56689 

66713 

56736 

66760 

66784 

56808 

568S2 

66856 

56880 

66904 

56928 

66962 

66976 

57000 

57024 

67047 

57071 

67096 

67119 

67143 

67167 

67191 

67216 

67238 

57262 

67286 

57310 

67334 

57358 


82904 

82887 

82871 

82865 

82839 

82822 

82806 

82790 

82773 

82757 

82741 

82724 

82708 

82692 

82675 

82659 

82643 

82626 

82610 

82593 

82577 

82661 

82644 

82528 

82511 

82496 

82478 

82462 

82446 

82429 

82413 

82396 

82380 

82363 

82347 

82330 

82314 

82297 

82281 

82264 

82248 

82231 

82214 

82198 

82181 

82165 

82148 

82132 

82115 

82098 

82082 

82065 

82048 

82032 

82015 

81999 

81982 

81965 

81949 

81932 

81915 


N.  cos.  N.sine 


60 
59 
58 
57 
56 
55 
54 
63 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 
9 
S 
7 
6 
5 
4 
3 
2 
1 
0 


55  Degrees. 


56 


Log.  Sines  and  Tangents.    (35°)    Natural  Sines. 


TABLE  II. 


Sine. 

768591 
758772 
758952 
759132 
759312 
759492 
769672 
759852 
760031 
760211 
760390 
760569 
760748 
760927 
761106 
761285 
761464 
761642 
761821 
761999 
762177 
762356 
762534 
762712 
762889 
763067 
763245 
763422 
763600 
763777 
763954 
764131 
764308 
764485 
764662 
764838 
765015 
765191 
766367 
765544 
766720 

9.765896 
766072 
766247 
766423 
766598 
766774 
766949 
767124 
767300 
767475 

9.767649 
767824 
767999 
768173 
768348 
768522 
768697 
768871 
769045 
769219 


Cosine. 


D.  lU' 


30.1 
30.0 
30.0 
30.0 
30.0 
30.0 
29.9 
29.9 
29.9 
29.9 
29.9 
29.8 
29.8 
29.8 
29.8 
29.8 
29.8 
29.7 
29.7 
29.7 
29.7 
29.7 
29.6 
29.6 
29.6 
29.6 
29.6 
29.6 
29.5 
29.6 
29.6 
29.5 
29.5 
29.4 
29.4 
29.4 
29.4 
29.4 
29.4 
29.3 
29.3 
29.3 
29.3 
29.3 
29.3 
29.2 
29.2 
29.2 
29.2 
29.2 
29.1 
29.1 
29.1 
29.1 
29.1 
29.0 
29.0 
29.0 
29.0 
29.0 


Cosine. 

.913365 
913276 
913187 
913099 
913010 
912922 
912833 
912744 
912665 
912566 
912477 

.912388 
912299 
912210 
912121 
912031 
911942 
911853 
911763 
9U674 
911584 

.911496 
911405 
911315 
911226 
911136 
911046 
910956 
910866 
910776 
910686 

.910596 
910506 
910415 
910326 
910235 
910144 
910054 
909963 
909873 
909782 

.909691 
909601 
909510 
909419 
909328 
909237 
909146 
909055 
908964 
908873 

.908781 
908690 
9086^9 
908607 
908416 
908324 
908233 
908141 
908049 
907958 


Sine. 


D.  10' 


14.7 

14.7 

14.8 

14.8 

14.8 

14.8 

14.8 

14.8 

14.8 

14.8 

14.8 

14.8 

14.9 

14.9 

14.9 

14.9 

14.9 

14.9 

14.9 

14.9 

14.9 

14,9 

14.9 

15.0 

16.0 

16.0 

15.0 

15.0 

15.0 

15.0 

16.0  Q 

15.0^ 

16.0 

16.0 

15.1 

16.1 

15.1 

15.1 

15.1 

16.1 

16.1 

15.1 

16.1 

16.1 

15.1 

16.2 

15.2 

16.2 

15.2 

15.2 

16.2 

16.2 

16.2 

15.2 

15.2 

16.3 

16.3 

15.3 

15.3 

15.3 


Tang. 


,846227 
846496 
845764 
846033 
846302 
846570 
846839 
847107 
847376 
847644 
847913 

.848181 
848449 
848717 


849254 
849522 
849790 
860058 
850326 
860593 
860861 
851129 
861396 
851664 
851931 
852199 
852466 
852733 
863001 
863268 
853636 
853802 
854069 
864336 
864603 
854870 
865137 
866404 
866671 
866938 
856204 
856471 
856737 
857004 
867270 
857637 
857803 
858069 
858336 
858602 
.868868 
859134 
859400 
859666 
859932 
860198 
860464 
860730 
860995 
861261 


Cotang. 


Cotang.     I  N.  sine.  N.  cos 


10, 


10 


10 


10 


10 


10 


164773 
154504 
154236 
163967 
163698 
153430 
153161 
152893 
152624 
152366 
152087 
161819 
151551 
161283 
151014 
150746 
150478 
150210 
149942 
149676 
149407 
149139 
148871 
148604 
148336 
148069 
147801 
147534 
147267 
146999 
146732 
146466 
146198 
145931 
145664 
145397 
145130 
144863 
144696 
144329 
144062 
143796 
143629 
143263 
142996 
142730 
142463 
142197 
141931 
141664 
141398 
141132 
140866 
140600 
140334 
140068 
139802 
139536 
139270 
139005 
138739 


67358 

57381 

57405 

57429 

57453 

: 57477 

57501 

: 57524 

57648 

67572 

57596 

57619 

I  57643 

I  57667 

i  57691 

I  57715 

57738 

1 67762 


81916 
81899 
81882 
81866 
81848 
81832 
81815 
81798 
81782 
81765 
81748 
81731 
81714 
81698 
81681 
81664 
81647 
81631 


57786|816]4 
57810:81597 
57833[81580 
i  57867181563 
67881 181546 
57904|81530 
57928:81513 
57962 '81496 
57976|81479 
67999181462 
58023!81445 
58047 '81428 
|58070;81412 
j  58094  81396 
6811881378 


'58141 
I  58165 
I  58189 
58212 
58236 
58260 
68283 


81361 
81344 
81327 
81310 
81293 
81276 
81259 


58307  81242 
58330  81226 
68354  81208 
68378  81191 
6840181174 


Tang. 


58425 
58449 
68472 
58496 
68519 
58543 
I  58567 
I  58590 
58614 
58637 
58661 
68684 
58708 
58731 


81157 
81140 
81123 
81106 
81089 
81072 
81055 
81038 
81021 
81004 
80987 
80970 
.S0953 
80^36 


58765  80919 
58779  809U2 


N.  cos.  N.sine. 


54  Degrees. 


TABLE  II. 


Log.  Sines  and  Tangents.    (36°)    Natural  Sines. 


57 


Sine. 


9.769219 
769393 
769666 
769740 
769913 
770087 
770260 
770433 
770606 
770779 
770952 

9.771125 
771298 
771470 
771643 
771815 
771987 
772169 
772331 
772503 
772675 

9.772847 
773018 
773190 
773361 
773533 
773704 
773876 
774046 
774217 
774388 

9.774568 
774729 
774899 
776070 
775240 
775410 
775580 
775760 
775920 
776090 
776259 
776429 
776598 
776768 
776937 
777106 
777276 
777444 
777613 
777781 
777960 
778119 
778287 
778456 
778624 
778792 
778960 
779128 
779295 
779463 


Cosine. 


D.  10" 

29.0 
28.9 
28.9 
28.9 
28.9 
28.9 
28.8 
28.8 
28.8 
28.8 
28.8 
28.8 
28.7 
28.7 
28.7 
28.7 
28.7 
28.7 
28.6 
28.6 
28.6 
28.6 
28.6 
28.6 
28.5 
28.5 
28.6 
28.5 
28.5 
28.5 
28.4 
28.4 
28.4 
28.4 
28.4 
28.4 
28.3 
28.3 
28.3 
28.3 
28.3 
28.3 
28.2 
28.2 
28.2 
28.2 
28.2 
28.1 
28.1 
28.1 
28.1 
28.1 
28.1 
28.0 
28.0 
28.0 
28.0 
28.0 
28.0 
27.9 


9. 


Cosine. 

.907968 
907866 
907774 
907682 
907690 
907498 
907406 
907314 
907222 
907129 
907037 

.906945 
906862 
906760 
906667 
906576 
906482 
906389 
906296 
906204 
906111 
906018 
906925 
906832 
906739 
906645 
906662 
905459 
906366 
905272 
906179 

.905085 
904992 
904898 
904804 
904711 
904617 
904523 
904429 
904335 
904241 

.904147 
904053 
903969 
903864 
903770 
903676 
903581 
903487 
903392 
903298 

.903202 
903108 
903014 
902919 
902824 
902729 
902634 
902539 
902444 
902349 


Sine. 


D.  10" 


15.3 
15.3 
16.3 
15.3 
16.3 
16.3 
15.3 
16.4 
15.4 
16.4 
16.4 
16.4 
15.4 
16.4 
15.4 
15.4 
16.4 
15.5 
15.5 
15.6 
16.5 
15.5 
15.5 
15.6 
15.6 
15.5 
15.5 
15.5 
15.6 
15.6 
15.6 
15.6 
15.6 
16.6 
15.6 
16.6 
16.6 
15.6 
15.7 
16.7 
15.7 
15.7 
15.7 
15.7 
16.7 
16.7 
16.7 
16.7 
15.7 
15.8 
16.8 
16.8 
16.8 
15.8 
15.8 
15.8 
15.8 
16.8 
16.9 
15.9 


Tang, 


.861261 
861527 
861792 
862058 
862323 
862589 
862854 
863119 
863385 
863660 
863916 

.864180 
864446 
864710 
864976 
866240 
866606 
866770 
866036 
866300 
866564 

.866829 
867094 
867368 
867623 
867887 
868152 
868416 
868680 
868945 
869209 

.869473 
869737 
870001 
870265 
870529 
870793 
871057 
871321 
871586 
871849 

.872112 
872376 
872640 
872903 
873167 
873430 
873694 
873967 
874220 
874484 

.874747 
876010 
876273 
875536 
875800 
876063 
876326 
876589 
876851 
877114 


Cotang. 


D.  10" 


44.3 
44.3 
44.2 
44.2 
44.2 
44.2 
44.2 
44.2 
44.2 
44.2 
44.2 
44.2 
44.2 
44.2 
44.1 
44.1 
44.1 
44.1 
44.1 
44.1 
44.1 
44.1 
44.1 
44.1 
44.1 
44.1 
44.0 
44.0 
44.0 
44.0 
44.0 
44.0 
44.0 
44.0 
44.0 
44.0 
44.0 
44.0 
44.0 
44.0 
43.9 
43.9 
43.9 
43.9 
43.9 
43.9 
43.9 
43.9 
43.9 
43.9 
43.9 
43.9 
43.9 
43.8 
43.8 
43.8 
43.8 
43.8 
43.8 
43.8 


Cotang.     I  N.  sine.  N.  cos . 


10.138739 
138473 
138208 
137942 
137677 
137411 
137146 
136881 
136615 
136350 
136085 

10.135820 
135656 
136290 
135025 
134760 
134495 
134230 
133966 
133700 
133436 

10.133171 
132906 
132642 
132377 
132113 
131848 
131684 
131320 
131055 
130791 

10.130527 
130263 
129999 
129736 
129471 
129207 
128943 
128679 
128416 
128151 

10.127888 
127624 
127360 
127097 
126833 
126570 
126306 
126043 
126780 
125516 

10.125253 
124990 
124727 
124464 
124200 
123937 
123674 
123411 
123149 
122886 


58779  80902 
58802  80885 
5882680867 
5884980860 
68873:80833 
58896  80816 
68920;80799 
58943;80782 
68967  80765 
58990,80748 
59014'80730 
59037:80713 
69061  80696 


59084 
59108 
69131 
59154 
69178 
59201 
59226 
69248 
69272 
69296 
59318 
69342 
69366 
69389 
59412 
69436 
59459 


80679 
80662 
80644 
80627 
80610 
80593 
80576 
80568 
80541 
80524 
80507 
80489 
80472 
80456 
80438 
80422 
80403 


69482180386 
59506180368 
59529180351 
59552180334 
5957680316 
59599  80299 


59622 
59646 
69669 
59693 
59716 
59739 


80282 
80264 
80247 
80230 
80212 
80196 


59763180178 
69786180160 
59803  80143 
59832 18U126 
5985(i  180108 
59879|800yi 
59902  80073 
5992G  80056 
69949 180038 
699;2!80021 


59995 
60019 
60042 
60065 
60089 
60112 
60135 
60158 
60182 


Tang.   II  N.  cos.  N.sine. 


80003 
79986 
79968 
79951 
79934 
79916 
79899 
79881 
79864 


53  Degrees. 


58 


Log.  Sines  and  Tangents.    (37°)    Natural  Sines. 


TABLE  II. 


D.  10" 


0 

1 
2 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
20 
39 
31 
32 
33 
34 
35 
36 
37 


40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


Sine. 

,779463 
779631 
779798 
779966 
780133 
780300 
780467 
780634 
780801 
780968 
781134 
781301 
781468 
781634 
781800 
781966 
782132 
782298 
782464 
782630 
782796 
782961 
783127 
783282 
783458 
783623 
783788 
783953 
784118 
784282 
784447 
784612 
784776 
784941 
785105 
785269 
785433 
785597 
785761 
785925 
786089 
786252 
786416 
786579 
786742 
786906 
787069 
787232 
787395 
787657 
787720 
787883 
788045 
788208 
788370 
788532 
788694 
788856 
789018 
789180 
789342 


Cosine. 


27.9 
27.9 
27.9 
27.9 
27.9 
27.8 
27.8 
27.8 
27.8 
27.8 
27.8 
27.7 
27.7 
27.7 
27.7 
27.7 
27.7 
27.6 
27.6 
27.6 
27.6 
27.6 
27.6 
27.5 
27.5 
27.5 
27.5 
27.5 
27.5 
27.4 
27.4 
27.4 
27.4 
27.4 
27.4 
27.3 
27.3 
27.3 
27.3 
27.3 
27.3 
27.2 
27.2 
27.2 
27.2 
27.2 
27.2 
27.1 
27.1 
27.1 
27.1 
27.1 
27.1 
27.1 
27.0 
27.0 
27.0 
27.0 
27.0 
27.0 


Cosine.  |D.  10" 


.902349 
902263 
902168 
902063 
901967 
901872 
901776 
901681 
901686 
901490 
901394 

.901298 
901202 
901106 
901010 
900914 
900818 
900722 
900626 
900529 
900433 

.900337 
900242 
900144 
900047 
899961 
899854 
899757 
899660 
899664 
899467 

.899370 
899273 
899176 
899078 


898884 
898787 
898689 
898592 
898494 
.898397 
898299 
898202 
898104 
898006 
897908 
897810 
897712 
897614 
897516 
.897418 
897320 
897222 
897123 
897025 
896926 
896828 
896729 
896631 
896532 


Sine. 


15.9 
15.9 
15.9 
15  9 
15  9 
15.9 
15  9 
15  9 
15.9 

15  9 
16,0 

16  0 
16  0 
16  0 
16,0 
16  0 
16  0 
16'0 
16  0 
16*0 
16'l 
16  1 


16 

16 

16 

16 

16 

16 

16 

16 

16 

16,2 

16.2 

16.2 

16.2 

16.2 

16.2 

16.2 

16,2 

16.2 

16.3 

16.3 

16.3 

16.3 

16.3 

16.3 

16.3 

16.3 

16.3 

16.3 

16.3 

16.4 

16.4 

16.4 

16.4 

16.4 

16.4 

16.4 

16.4 

16.4 


Tang. 


9. 


9. 


.877114 
877377 
877640 
877903 
878166 
878428 
878691 
878953 
879216 
879478 
879741 
880003 
880265 
880528 
880790 
881062 
881314 
881576 
881839 
882101 
882363 
882625 
882887 
883148 
883410 
883672 
883934 
884196 
884467 
884719 
884980 
886242 
885503 
885766 
886026 
886288 
886549 
886810 
887072 
887333 
887594 
887855 
888116 
888377 
888639 
888900 
889160 
889421 
889682 
889943 
890204 
890465 
890725 
890986 
891247 
891507 
891768 
892028 
892289 
892649 
892810 


D.  10' 


Cotang. 


43.8 
43.8 
43.8 
43.8 
43.8 
43.8 
43.8 
43.7 
43.7 
43.7 
43.7 
43.7 
43.7 
43.7 
43.7 
43.7 
43.7 
43,7 
43.7 
43.7 
43.6 
43,6 
43.6 
43.6 
43.6 
43.6 
43.6 
43.6 
43.6 
43.6 
43.6 
43.6 
43.6 
43.6 
43.6 
43.6 
43.5 
43.5 
43.5 
43.5 
43.5 
43.5 
43.6 
43.5 
43.5 
43.5 
43,5 
43.5 
43.5 
43.5 
43.4 
43.4 
43.4 
43.4 
43.4 
43.4 
43.4 
43.4 
43.4 
43.4 


Cotang.  [|N.sine. 


10.122886 
122623 
122360 
122097 
121835 
121672 
121309 
121047 
120784 
120522 
120259 

10.119997 
119736 
119472 
119210 
118948 
118686 
118424 
118161 
117899 
117637 

10.117375 
117113 
116852 
116590 
116328 
116066 
116804 
115543 
115281 
115020 

10.114758 
114497 
114236 
113974 
113712 
113451 
113190 
112928 
112667 
112406 

10.112145 
111884 
111623 
111361 
111100 
110840 
110679 
110318 
110067 
109796 

10.109635 
109275 
109014 
108763 
108493 
108232 
107972 
107711 
107451 
107190 


60182 
60206 
60228 
60251 
60274 
60298 
60321 
60344 
60367 
60390 
60414 
60437 
60460 
60483 
G0606 
60529 
60653 
60576 
60599 
60622 
60645 
60668 
60691 
60714 
60738 
60761 
60784 
60807 
60830 
60853 
60876 
60899 
60922 
60946 
60968 
60991 
61016 
61038 
61061 
61084 
61107 
61130 
61163 
61176 
61199 
61222 
61246 
61268 
61291 
61314 
61337 
61360 
61383 


N.  cos, 


9864 
79846 
79829 
79811 
79793 
79776 
79758 
79741 
79723 
79706 
79688 
79671 
79668 
79635 
79618 
79600 
79583 
79565 
79547 
79530 
79512 
79494 
79477 
79459 
79441 
79424 
79406 
79388 
79371 
79363 
79336 
79318 
79300 
79282 
79264 
79247 
79229 
79211 
79193 
79176 
79168 
79140 
79122 
79106 
79087 
79069 
79061 
79033 
79016 
78998 

8980 
78962 
78944 


61406  78926 
61429  78908 


61451 
; 61474 
161497 
1 61520 
I  61643 
i  61666 


Tang. 


78891 
78873 
78865 
78837 
78819 
78801 
N.  COS.  N.sine. 


52  Degrees. 


TABLE  n. 


liOg.  Sines  and  Tangents.    (38°)    Natural  Sines. 


59 


0 
1 
2 
3 
4 
6 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
3S( 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


Sine. 


D.  10" 


9.789342 
789504 
789665 
789827 
789988 
790149 
790310 
790471 
790632 
790793 
790954 

9.791115 
791276 
791436 
791696 
791757 
791917 
792077 
792237 
792397 
792557 

9.792716 
792876 
793035 
793195 
793354 
793514 
793673 
793832 
793991 
794150 

9.794308 
7944671 
794626  I 
794784 ; 
794942  I 
795101 ' 
795259 
795417 
795575 
795733 

9.795891 
796049 
796206 
79G364 
796521 
796679 
795836 
796993 
797150 
797307 

9.797464 
797621 
797777 
797934 
798091 
798247 
798403 
79S560 
798716 
798872 


26.9 
26.9 
26.9 
26.9 
26.9 
26.9 
26.8 
26.8 
26.8 
26.8 
26.8 
26.8 
26.7 


9. 


26.7 
26.7 
26.7 
26.7 
26.7 
26.6 
26.6 
26.6 
26.6 
26.6 
26.6 
26.5 
.6 
.5 
.5 
.6 
.6 
.4 
.4 
.4 
.4 


26 


26 

26 

26 

26 

26 

26 

26 

26.4 

26.4 

26.4 

26.3 

26.3 

26.3 

26.3 

26.3 

26.3 

26.3 

26.2 

26.2 

26.2 

26 

26 

26 

26 

26 

26. 1 

26.1 

26.1 

26-1 

26.1 

26.0 

26.0 

26.0 


Cosine. 


D.  10' 


Cosine. 


896532 
896433 
896335 
896236 
896137 
896038 
895939 
895840 
895741 
895641 
895542 
.895443 
895343 
895244 
895145 
895045 
894945 
894846 
894746 
894646 
894546 
.894446 
894346 
894246 
894146 
894046 
893946 
893846 
893745 
893645 
893544 
.893444 
893343 
803243 
893142 
893041 
892940 
892839 
892739 
892638 
892536 
.892436 
892334 
892233 
892132 
892030 
891929 
891827 
891726 
891624 
891623 
.891421 
891319 
89121/ 
891116 
891013 
890911 
890809 
8907U7 
890605 
890503 


Sine. 


Tanf 


D.  10' 


16.4 
16.5 
16.5 
16.5 
16.5 
16.5 
16.5 
16.5 
16.5 
16.5 
16.5 
16.6 
16.6 
16.6 
16.6 
16.6 
16.6 
16.6 
16.6 
16.6 
16.6 
16.7 
16.7 
16.7 
16.7 
16.7 
16.7 
16.7 
16.7 
16.7 
16.7 
16.8 
16.8 
16.8 
16.8 
16.8 
16.8 
16.8 
16.8 
16.8 
16.8 
16.9 
16.9 
16.9 
16.9 
16.9 
16.9 
16.9 
16.9 
16.9 
17.0 
17.0 
17.0 
17.0 
17.0 
17.0 
17.0 
17.0 
17.0 
17.0 


9. 


892810 
893070 
893331 
893591 
893851 
894111 
894371 
894632 
894892 
895152 
895412 
895672 
895932 
896192 
896452 
896712 
896971 
897231 
897491 
897761 
898010 
898270 
898630 
898789 
899049 
899308 
899668 


900086 
900346 
900605 
.900864 
901124 
901383 
901642 
901901 
902160 
902419 
902679 
902938 
903197 
.903466 
903714 
903973 
904232 
904491 
904760 
905008 
906267 
905626 
905784 
.906043 
906302 
906660 
906819 
907077 
907336 
907694 
907862 
908111 
908369 
Cotang. 


43.4 

43.4 

43.4 

43.4 

43.4 

43.4 

43.4 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.2 

43 

43 

43 

43 

43 

43 

43 

43 

43 

43.2 

43.2 

43 

43 

43 

43 

43 

43 

43 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.0 


Cotang.  I N.  sine.[N.  cos 


10, 


10. 


10 


10 


10 


10 


1071901 
106930 1 
106669 
106409 
106149 
106889 
105629 
105368 
105108 
104848 
104688 
104328 
104068 
103808 
103548 
103288 
103029 
102769 
102509 
102249 
101990 
101730 
101470 
101211 
100951 
100692 
100432 
100173 
099914 
099654 
099395 
099136 
098876 
098617 
098358 
098099 
097840 
097581 
097321 
097062 
096803 
09G545 
096286 
096027 
095768 
095509 
096260 
094992 
094733 
094474 
094216 
093957 
093698 
093440 
093181 
092923 
092664 
092406 
092148 
091889 
091631 


Tang. 


61566 
61689 
61612 
61635 
61668 
61681 
61704 
61726 
61749 
61772 
61796 
61818 
61841 
61864 
61887 
61909 
61932 
61955 
61978 
62001 
62024 
62046 
62069 
62092 
62115 
62138 
62160 
62183 
62206 
62229 
62251 
62274 
62297 
62320 
62342 
62366 
62388 
62411 
62433 
62466 
62479 
62602 
62524 
62547 
62570 
62592 
62615 
62638 
62660 
62683 
6270b 
62728 
62/51 
62774 
62796 
62819 
62842 
62864 
62887 
62909 
62932 


78801 
78783 

8766 
78747 
78729 

8711 
78694 
78676 
78668 
78640 
78622 
78604 
78586 
78568 

8550 
78532 
78514 
78496 
78478 
78460 
78442 
78424 
78406 
78387 
78369 
78351 
78333 
78315 
78297 

8279 
78261 

8243 
78225 

8206 

8188 
78170 
78152 
78134 

8116 

8098 
78079 
78061 
78043 
78025 
78007 
77988 
77970 
77952 
77934 
77916 
77897 
77879 
77861 
77843 
77824 
77806 
77788 
77769 

77751 
77733 
77715 


N.  COS.  N.sine 


51  Degrees. 


60 


Log.  Sines  and  Tangents,    (39°)    Natural  Sines.  TABLE  II. 


0 
1 
2 
3 

4 
5 
6 
7 
8 
9 
10 
Jl 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 


Sine. 


9.798772 
799028 
799184 
799339 
799495 
799651 
799806 
799962 
800117 
800272 
800427 
9.800582 
800737 
800892 
801047 
801201 
801356 
801611 
801665 
801819 
801973 
9.802128 
802282 
802436 
802589 
802743 
802897 
803050 
803204 
-^  803357 
30   803511 


D.  10" 


31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
61 
52 
53 
54 
56 
56 
57 
58 
59 
60 


9.803664 
803817 
803970 
804123 
804270 
804428 
804581 
804734 
804886 
805039 
9.805191 
805343 
805495 
805647 
805799 
805951 
806103 
806254 
806406 
806557 
9.806709 
806860 
807011 
807163 
807314 
807465 
807615 
807766 
807917 
808067 


Cosine. 


26.0 

26.0 

26.0 

25.9 

25.9 

25.9 

25.9 

25.9 

25.9 

25.8 

25.8 

25.8 

25.8 

25.8 

25.8 

25.8 

25.7 

25.7 

25.7 

25.7 

25.7 

25.7 

25.6 

25.6 

25.6 

25.6 

25.6 

25.6 

25.6 

25.5 

25.5 

25.5 

25.5 

25.5 

25.5 

25.4 

25.4 

26.4 

25.4 

25.4 

25.4 

26.4 

26.3 

25.3 

25.3 

25.3 

25.3 

25.3 

25.3 

25.2 

25.2 

25.2 

25.2 

25.2 

25.2 

25.2 

25.1 

25.1 

25.1 

25.1 


Cosine. 

1.890503 
890400 
890298 
890195 
890093 
889990 
889888 
889785 
889682 
889579 
889477 
.889374 
889271 
889168 
889084 
888961 


888766 
888651 
888548 
888444 
5.888341 
888237 
888134 
888030 
887926 
887822 
887718 
887614 
887510 
887406 
>. 887302 
887198 
887093 
886989 
88G885 
886780 
886676 
886571 
880466 
88G362 
'.886257 
886152 
886047 
885942 
885837 
885732 
886627 
885522 
885416 
885311 
.885205 
886100 
884994 
884889 
884783 
884677 
884572 
884466 
884360 
884264 


D.  10" 


Sine. 


17.0 

17.1 

17.1 

17.1 

17.1 

17.1 

17.1 

17.1 

17.1 

17.1 

17.1 

17.2 

17.2 

17.2 

17.2 

17.2 

17.2 

17.2 

17.2 

17.2 

17.3 

17.3 

17.3 

17.3 

17.3 

17.3 

17.3 

17.3 

17.3 

17.3 

17.4 

17.4 

17.4 

17.4 

17.4 

17.4 

17.4 

17.4 

17.4 

17.4 

17.5 

17.6 

17.5 

17.6 

17.5 

17.6 

17.6 

17.5 

17.5 

17.5 

17.6 

17.6 

17.6 

17.6 

17,6 

17.6 

17.6 

17.6 

17.6 

17.6 


Tang. 


9.908369 
908G28 
908886 
90J144 
909402 
909660 
909918 
910177 
910435 
910693 
910951 
9.911209 
911467 
911724 
911982 
912240 
912498 
912756 
913014 
913271 
913629 
). 913787 
914044 
914302 
914660 
914817 
915075 
915332 
915590 
915847 
916104 
1.911)362 
916619 
916877 
917134 
917391 
917648 
91/905 
918163 
918 120 
918677 
.918934 
919191 
919448 
919705 
919962 
920219 
920476 
920733 
920990 
921247 
.921503 
921760 
922017 
922274 
922530 
922787 
923044 
923300 
923557 
923813 


D.  10' 


Co  tang. 


43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.7 


Cotang. 


10 


10 


10 


10.091631 

091372 

091114 

090856 

090598 

090340 

090082 

089823 

089565 

089307 

089049 

088791 

088533 

088276 

088018 

087760 

087502 

087244 

086986 

086729 

086471 

.086213 

085956 

085698 

085440 

085183 

084925 

084668 

084410 

084153 

033896 

.083638 

083381 

083123 

082866 

082609 

082352 

082095 

081837 

081680 

081323 

081066 

080809 

080552 

080295 

080038 

079781 

0 J 9524 

079267 

079010 

0/8753 

.078497 

078240 

077983 

077726 

077470 

077213 , 

076956 : 

076700 

076443 

076187  i 


N.  sine.  N.  cos. 

77715 
77696 
77678 
77660 
77641 
77623 
7605 
77586 
77568 
77550 
77531 
77513 
77494 
77476 
77458 
77439 
77421 
77402 
77384 
77366 
77347 
77329 
77310 
77292 
77273 
77255 
77236 
77218 


62932 

62955- 

6297 

6300UI 

63022 

63045 

63068 

63090 

63113 

63135 

63158 

93180 

63203 

63225 

63248 

63271 

63293 

63316 

63338 

63361 

63383 

6340fj 

63428 

63461 

63473 

63496 

63518 

63540 

63563177199 

63585177181 


10 


10 


6360.^ 

63630 

6365Li 

63676 

63698 

i  63720 

ji6374- 

163765 

1163787 

I  j 63810 

!  I  63832 

1:63854 

!  16387. 

j|6ab99 

||6392l: 

i  1 63944 

|i6396{. 

||63y8iJ 

1164011 

164033 

64050 

i640?S 

64100 

64123 

' 64146 

64167 

64190 

64212 

64234 

64250 

; 64279 


77162 

77144 

77126 

77107 

77088 

77070 

77051 

77033 

77014 

76996 

76977 

76959 

7()940 

70921 

70903 

76884 

76866 

76847 

70828 

76810 

76791 

76772 

76754 

76735 

76717 

76698 

76679 

76661 

76642 

76623 

76004 


Tang.   I  N.  cok.  N.piue, 


47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 

0 


50  Degrees. 


TABLE  II. 


Log.  Sines  and  Tangents.    (40°)    Natural  Sines. 


61 


Cotang.  I  N  .sine.  N.  cos 


0 

1 

2 
3 
4 
6 
6 
7 
8 
9 
10 
.1 
12 
13 
14 
16 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
82 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


9. 


Sine. 


D.W 


.808067 
808218 


808519 
808669 
808819 


809119 
809269 
809419 
809669 
809718 
809868 
810017 
810167 
810316 
810465 
810614 
810763 
810912 
811061 
811210 
811358 
811507 
811656 
811804 
811962 
812100 
812248 
812396 
812544 
.812692 
812840 
812988 
813136 
81-3283 
813430 
813578 
813726 
813872 
814019 
.814166 
814313 
814460 
814607 
814753 
814900 
815046 
815193 
815339 
815486 
.816631 
815778 
816924 
816069 
816215 
816361 
816607 
816652 
816798 
816943 
Cosine. 


25.1 
25.1 
25.1 
26.0 
26.0 
25.0 
26.0 
26.0 
25.0 
24.9 
24.9 
24.9 
24.9 
24.9 
24.9 
24.8 
24.8 
24.8 
24.8 
24.8 
24.8 
24.8 
24.7 
24.7 
24.7 
24.7 
24.7 
24.7 
24.7 
24.6 
24.6 
24.6 
24.6 
24.6 
24.6 
24.6 
24.5 
24.6 
24.5 
24.6 
24.6 
24.6 
24.5 
24.4 
24.4 
24.4 
24.4 
24.4 
24.4 
24.4 
24.3 
24.3 
24.3 
24.3 
24.3 
24.3 
24.3 
24.2 
24.2 
24.2 


Cosir 


D.  10" 


,884254 
884148 
884042 
883936 
883829 
883723 
883617 
883610 
883404 
883297 
883191 
,883084 
882977 
882871 
882764 
882657 
882560 
882443 
882336 
882229 
882121 
,882014 
881907 
881799 
881692 
881584 
881477 
881369 
881261 
881153 
881046 
,880938 
880830 
880722 
880613 
880506 
880397 
880289 
880180 
880072 
879963 
.879865 
879746 
879637 
879529 
879420 
879311 
879202 
879093 
878984 
878876 
.878766 
878656 
878647 
878438 
878328 
878219 
878109 
877999 
877890 
877780 
Sine. 


17.7 

17.7 

17.7 

17.7 

17.7 

17.7 

17.7 

17.7 

17.7 

17.8 

17.8 

17.8 

17.8 

17.8 

17.8 

17.8 

17.8 

17.8 

17.9 

17.9 

17.9 

17.9 

17.9 

17.9 

17.9 

17.9 

17.9 

17.9 

18.0 

18.0 

18.0 

18.0 

18.0 

18.0 

18.0 

18.0 

18.0 

18. 

18. 

18. 

18. 

18. 

18. 

18. 

18. 

18. 

18. 

18.2 

18.2 

18.2 

18,2 

18.2 

18.2 

18.2 

18.2 

18.2 

18.3 

18.3 

18.3 

18.3 


Tang.   D.  10" 


.923813 
924070 
924327 
924583 
924840 


925352 
925609 
925866 
926122 
926378 

.926634 
926890 
927147 
927403 
927669 
927916 
928171 
928427 
928683 
928940 

.929196 
929462 
929708 
929964 
930220 
930475 
930731 
930987 
931243 
931499 

•931755 
932010 
932266 
932522 
932778 
933033 
933289 
933546 
933800 
934056 

.934311 
934567 
934823 
935078 
935333 
935589 
935844 
936100 
936355 
936610 

.936866 
937121 
937376 
937632 
937887 
938142 
938398 
938653 
938908 
939163 


Cotang. 


42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.6 
42.6 
42.6 
42.6 
42.6 
42.6 
42.6 
42.6 
42.6 
42.6 
42.6 
42.6 
42.6 
42.6 
42.6 
42.6 
42,6 
42.6 
42,6 
42.6 
42,6 
42,6 
42.6 
42.6 
42,6 
42.6 
42,6 
42,6 
42.5 
42.5 
42,5 
42,6 
42,5 
42,5 
42,5 
42.5 


10.076187 
076930 
075673 
076417 
076160 
074904 
074648 
074391 
074136 
073878 
073622 

10.073366 
073110 
072863 
072597 
072341 
072086 
071829 
071573 
071317 
071060 

10.070804 
070548 
070292 
070036 
069780 
069625 
069269 
069013 
068757 
068501 

10,068245 
067990 
067734 
067478 
067222 
066967 
066711 
066456 
066200 
065944 

10.066689 
065433 
065177 
064922 
064667 
064411 
064166 
063900 
063645 
063390 

10.063134 
002879 
062624 
062368 
062113 
061868 
061602 
061347 
061092 
060837 


64279 
64301 
64323 
64346 
64368 
64390 
64412 
64435 
64467 
64479 
64501 
64524 
64546 
64568 
64690 
64612 
64635 
64667 
64679 
64701 
64723 
64746 
64768 
64790 
64812 
64834 
64856 
64878 
64901 
64923 
64946 
64967 
64989 
66011 
65033 
66066 
65077 
65100 
66122 
65144 
65166 
65188 
65210 
65232 
66254 
65276 
65298 
65320 
65342 
G5364 
!  1 65386 
I  6540b 
i  1 65430 
65452 
65474 
65496 
65618 
66640 
65562 
G5584 
65006 


Tang. 


76604 
76586 
76667 
76548 
76530 
76611 
6492 
76473 
76455 
76436 
76417 
76398 
76380 
6361 
6342 
76323 
76304 
76286 
76267 
76248 
6229 
76210 
76192 
76173 
76154 
76135 
76116 
76097 
76078 
76059 
6041 
6022 
76003 
75984 
5965 
5946 
75927 
76908 
76889 
75870 
75851 
75832 
76813 
76794 
76775 
75756 
76738 
76719 
75700 
75680 
75661 
76642 
76623 
75604 
76585 
75666 
75647 
75528 
76509 
76490 
76471 
N,  COS.  N.sine. 


49  Degrees. 


62 


Log.  Sines  and  Tangents.    (41°)    Natural  Sines. 


TABLE  n. 


5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
36 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


Sine. 


816943 
817088 
817233 
817379 
817524 
817668 
817813 
817958 
818103 
818247 
818392 

9.818536 
818681 
818825 
818969 
819113 
819257 
819401 
819545 
819689 
819832 

9.819976 
820120 
820263 
820406 
820550 
820693 
820836 
820979 
821122 
821265 

9.821407 
821550 
821693 
821835 
821977 
822120 
822262 
822404 
822546 
822688 

9.822830 
822972 
823114 
823255 
823397 
823539 
823680 
823821 
823963 
824104 

9.824245 
824386 
824527 
824668 
824808 
824949 
825090 
825230 
825371 
826511 


D.  Ky 


Cosine. 


24.2 

24.2 

24.2 

24.2 

24.1 

24.1 

24.1 

24.1 

24.1 

24.1 

24 

24.0 

24.0 

24.0 

24.0 

24.0 

24.0 

24.0 

23.9 

23.9 

23.9 

23.9 

23.9 

23.9 

23.9 

23.8 

23.8 

23.8 

23.8 

23.8 

23.8 

23.8 

23.8 

23.7 

23.7 

23.7 

23.7 

23.7 

23.7 

23.7 

23.6 

23.6 

23.6 

23.6 

23.6 

23.6 

23.6 

23.5 

23.5 

23.6 

23.5 

23.5 

23.5 

23.5 

23.4 

23.4 

23.4 

23.4 

23.4 

23.4 


Cosine. 


876678 
.876568 
876457 
876347 
876236 
876125 
876014 
875904 
875793 
875682 
875571 
.875459 
875348 
876237 
875126 
875014 
874903 
874791 
874680 
874568 
874456 
.874344 
874232 
874121 
874009 
873896 
873784 
873672 
873560 
873448 
873335 
,873223 
873110 
872998 
872885 
872772 
872659 
872547 
872434 
872321 
872208 
872095 
871981 
871868 
871765 
871641 
871528 
871414 
871301 
871187 
871073 


D.  10" 


18.3 
18.3 
18.3 
18.3 
18.3 
18.4 
18.4 
18.4 
18.4 
18.4 
18.4 


Sine. 


18.4 
18.4 
18.4 
18.5 
18.5 
18.5 
18.5 
18.5 
18.5 
18.5 
18.5 
18.5 
18.5 
18.6 
18.6 
18.6 
18.6 
18.6 
18.6 
18.6 
18.6 
18.7 
18.7 
18.7 
18.7 
18.7 
18.7 
18.7 
18.7 
18.7 
18.7 
18.8 
18.8 
18.8 
18.8 
18.8 
18.8 
18.8 
18.8 
18.8 
18.9 
18.9 
18.9 
18.9 
18.9 
18.9 
18.9 
18.9 
18.9 


Tang. 


9.939163 
939418 
939673 
939928 
940183 
940438 
940694 
940949 
941204 
941458 
941714 

9.941968 
942223 
942478 
942733 
942988 
943243 
943498 
943752 
944007 
944262 

5.944517 
944771 
945026 
945281 
946535 
945790 
946045 
946299 
946654 
946808 

J.9470t>3 
947318 
947572 
947826 
948081 
948336 
948590 
948844 
949099 
949353 

).  949607 
949862 
950116 
950370 
960625 
950879 
951133 
951388 
961642 
951896 

).  962150 
952405 
952669 
962913 
953167 
963421 
953675 
953929 
954183 
954437  j 

Cotang.  I 


D.  10' 


42.5 

42.5 

42.5 

42.5 

42.5 

42.5 

42.5 

42.5 

42.5 

42 

42 

42 

42 

42 

42 

42 

42 

42 

42.5 

42.5 


42 

42 

42 

42 

42 

42 

42 

42 

42 

42 

42 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 


Cotang.  I  N.  sine 


42 

42 

42 

42 

42 

42 

42 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.4 

42.3 

42.3 

42.3 

42.3 

42.3 


65759 
65781 


10.060837  1 165606 
060582  1165628 
060327  1 165650 
060072  I '65672 
059817!]  65094 
059562  1 1  65716 
069306^165738 
069051 
058796 

058542!!  65803 
068286  I  65825 

10.058032  I  65847 
057777!  I  65869 
057522!!  65891 
057267  i 65913 
057012  ji  65935 
056757!  [65956 
056502  65978 
056248  66000 


N.  cos 


056993 
065738 
10.055483 
055229 


66022 
66044 
! 66066 
1 66088 


0549741  66109 
0547191  66131 
054466  I!  66163 
0542101166175 
053955 j  166197 
053701  166218 
053446  1 1 66240 
053192!  166262 

10.052937'  66284 
052682  66306 
052428  66327 
062174  66349 
0619191  66371 
0516641  66393 
061410  i 66414 
061156  66436 
060901]  1 66458 
060647]  66480 

10.050393!!  66501 
060138!  166523 
049884  !j  66546 
049630  ,66666 
049375!  1 66588 
049121  !!6661U 
048867  I  66632 
048612;  66653 
048358  66675 
048104;  66697 

10.047850  66718 
047696;  6674U 
047341:  66762 
0470871  66783 
0468331  66806 
046579  66827 
046325:  66848 
046071!  66870 
045817  I  j  66891 
0465631  66913 


76471 
75452 
76433 
75414 
75395 
75375 
75356 
75337 
75318 
76299 
76280 
76261 
76241 
76222 
5203 
76184 
75165 
75146 
75126 
75107 
75088 
75069 
76050 
75030 
75011 
74992 
74973 
74953 
74934 
74915 
74896 
74876 
74857 
74838 
74818 
74799 
74780 
74760 
74741 
74722 
74703 
74683 
74663 
74644 
74625 
74606 
74586 
74567 
74648 
74522 
74509 
74489 
74470 
74451 
74431 
74412 
74392 
74373 
74353 
74334 
74314 


Tang. 


N.  COS.  N.sine. 


48  Degrees. 


TABLE  II. 


Log.  Sines  and  Tangents.    (42°)    Natural  Sines. 


63 


9.826511 
825651 
825791 
825931 
826071 
826211 
826361 
826491 
826631 
826770 
826910 

9.827049 
827189 
827328 
827467 
827606 
827746 
827884 
828023 
828162 
828301 

9.828439 
828578 
828716 
828856 


Sine. 


829131 
829269 
829407 
829545 
829683 

9.829821 
829959 
830097 
830234 
830372 
830509 
830646 
830784 
830921 
831058 

9.831195 
831332 
831469 
831606 
831742 
831879 
832015 
832152 
832288 
832425 

9,832561 
832697 
832833 
832969 
833105 
833241 
833377 
833512 
833648 
833783 


D.  10"  Cosine. 


Cosine. 


23.4 
23.3 
23.3 
23. 3 
23.3 
23.3 
23.3 
23.3 
23.3 
23.2 
23.2 
23.2 
23.2 
23.2 
23.2 
23.2 
23.2 
23.1 
23.1 
23.1 
23.1 
23.1 
23.1 
23.1 
23.0 
23.0 
23.0 
23.0 
23.0 
23.0 
23.0 
22.9 
22.9 
22.9 
22.9 
22.9 
22.9 
22.9 
22.9 
22.8 
22.8 
22.8 
22.8 
22.8 
22.8 
22.8 
22.8 
22.7 
22.7 
22.7 
22.7 
22.7 
22.7 
22.7 
22.6 
22.6 
22.6 
22.6 
22.6 
22.6 


.871073 
870960 
870846 
870732 
870618 
870504 
870390 
870276 
870161 
870047 
869933 
.869818 
869704 
869589 
869474 
869360 
869245 
869130 
869015 
868900 
868785 
.868670 
868565 
868440 
868324 
868209 
868093 
867978 
867862 
867747 
867631 
.867515 
867399 
867283 
867167 
867051 


866819 
866703 
866586 
866470 
.866353 
866237 
866120 
866004 
866887 
866770 
866663 
866636 
866419 
866302 
.865185 
865068 
864950 
864833 
864716 
864698 
864481 
864363 
864246 
864127 


Sine. 


D.  10' 


19.0 
19.0 
19.0 
19.0 
19.0 
19.0 
19.0 
19.0 
19.0 
19.1 
19.1 
19.1 
19.1 
19.1 
19.1 
19.1 
19.1 
19.1 
19.2 
19.2 
19.2 
19.2 
19.2 
19.2 
19.2 
19.2 
19.2 
19.3 
19.3 
19.3 
19.3 
19.3 
19.3 
19.3 
19.3 
19.3 
19.4 
19.4 
19.4 
19.4 
19.4 
19.4 
19.4 
19.4 
19.6 
19.5 
19.5 
19.5 
19.5 
19.5 
19.5 
19.6 
19.6 
19.5 
19.6 
19.6 
19.6 
19.6 
19.6 
19.6 


Tang. 


9. 


9. 


9. 


954437 
964691 
964946 
966200 
965464 
965707 
956961 
956215 
966469 
956723 
966977 
967231 
957485 
957739 
957993 
958246 
958600 
958754 
959008 
969262 
959516 
959769 
960023 
960277 
960531 
960784 
961038 
961291 
961546 
961799 
962062 
962306 
962560 
962813 
963067 
963320 
963574 
963827 
964081 
964336 
964588 
964842 
965095 
965349 
966602 
966856 
966109 
966362 
966616 


967123 
.967376 
967629 
967883 
968136 
968389 
968643 
968896 
969149 
969403 
969656 


Cotang. 


D.  10" 


42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42 

42 

42 

42 

42 

42 

42 

42 

42 


Cotang. 


10. 


10 


10 


10 


10 


045663 
046309 
046065 
044800 
044546 
044293 
044039 
043786 
043631 
043277 
043023 
042769 
042516 
042261 
042007 
041764 
041600 
041246 
040992 
040738 
040484 
040231 
039977 
039723 
039469 
039216 
038962 
038709 
038455 
038201 
037948 
037694 
037440 
037187 
036933 
036680 
036426 
036173 
035919 
035665 
035412 
035158 
034905 
034651 
034398 
034145 
033891 


10 


033384 
033131 
032877 
.032624 
032371 
032117 
031864 
031611 
031357 
031104 
030861 
030697 
030344 


Tang. 


N.  sine.  N.  cos 


66913 
66935 
66956 
66978 
66999 
67021 
67043 
67064 
67086 
67107 
67129 
67151 
67172 
67194 
67215 
67237 
67258 
67280 
67301 
67323 
67344 
67366 
67387 
67409 
67430 
67462 
67473 
67496 
67616 
67638 
67559 
67680 
67602 
67628 
67645 
67666 
67688 
67709 
67730 
67752 
67773 
67795 
67816 
67837 
67859 
67880 
67901 
67923 
67944 
67965 
67987 
68008 
68029 
68051 
68072 
68093 
68116 
68136 
68157 
68179 
68200 


74314 
74295 
74276 
74256 
74237 
74217 
74198 
74178 
74159 
74139 
74120 
74100 
74080 
74061 
74041 
74022 
74002 
73983 
73963 
73944 
73924 
73904 
73886 
73865 
73846 
73826 
73806 
73787 
73767 
73747 
73728 
73708 
73688 
73669 
73649 
73629 
73610 
73590 
73570 
73661 
73631 
73611 
73491 
73472 
73462 
73432 
73413 
73393 
73373 
73363 
73333 
73314 
73294 
73274 
73254 
73234 
73215 
73195 
73175 
73155 
73135 
N.  cos.  N.sine. 


47  Degrees. 


64 


Log.  Sines  and  Tangents.  (43°)  Natural  Sines. 


TABLE  n. 


Sine. 


Cosine. 


9.864127 

864010 

863892 

863774 

863656 

863538 

863419 

863301 

863183 

863064 

862946 

9,862827 

862709 

862590 

862471 

862363 

862234 

862115 

861996 

861877 

861758 

9.861638 

861519 

861400 

861280 

861161 

861041 

860922 

860802 

860682 

860562 

9.860442 

860322 

860202 

860082 

859962 

859842 

859721 

859601 

859480 

859360 

9  859239 

859119 

858998 

858877 

858756 

858635 

858514 

858393 

858272 

^.  _   858151 

HJ'X  9.858029 

857908 

857786 

857665 

857543 

857422 

857300 

857178 

857056 

856934 


D.  10' 


21.9 
21.9 
21.9 
21.9 
21.8 
21.8 
21.8 
21.8 
21.8 


Sine. 


19.6 

19.6 

19.7 

19.7 

19.7 

19.7 

19.7 

19.7 

19.7 

19.7 

19.8 

19.8 

19.8 

19.8 

19.8 

19.8 

19.8 

19.8 

19.8 

19.8 

19.9 

19.9 

19.9 

19.9 

19.9 

19.9 

19.9 

19.9 

19.9 

20.0 

20.0 

20.0 

20.0 

20.0 

20.0 

20.0 

20.0 

20.1 

20.1 

20.1 

20.1 

20.1 

20.1 

20.1 

20.1 

20.2 

20.2 

20.2 

20.2 

20.2 

20.2 

20.2 

20.2 

20.2 

20.3 

20.3 

20.3 

20.3 

20.3 

20.3 


Tan-; 


.969656 
969909 
970162 
970416 
970669 
970922 
971175 
971429 
971682 
971935 
972188 
972441 
972694 
972948 
973201 
973454 
973707 
973960 
974213 
974466 
974719 

9.974973 
975226 
975479 
975732 
975985 
976238 
976491 
976744 
976997 
977250 

9.977503 
977756 
978009 
978262 
978515 
978768 
979021 
979274 
979527 
979780 

9.980033 


D.  10" 


980538 
980791 
981044 
981297 
981550 
981803 
982056 
982309 
.982562 
982814 
983067 
983320 
983573 
983826 
984079 
984331 
984584 
984837 


Cotang. 


42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 


Cotang.  iJN  .sine.  N.  cos. 


10.030344 
030091 
029838 
029584 
029331 
029078 
028825 
028571 
028318 
028065 
027812 

10.027559 
027306 
027052 
026799 
026546 
026293 
026040 
025787 
025634 
025281 

10.026027 
024774 
024521 
024268 
024015 
023762 
023509 
023256 
023003 
022750 

10.022497 
022244 
021991 
021738 
021485 
021232 
020979 
020726 
020473 
020220 

10.019967 
019714 
019462 
019209 
018956 
018703 
018460 
018197 
017944 
017691 

10.0174381 
017186  I 
016933  I 

016680 : 

016427 
016174 
015921 i 
015669  i 
016416 
015163 


I ! 68200 
168221 
:  1 68242 
'68264 
68285 
68306 
68327 
68349 


73135 
73116 
73096 
73076 
/3056 
73036 
73016 
72996 
68370  72976 


68391 
68412 
68434 


72957 
72937 
72917 


68455172897 


68476 
68497 
68518 
68539 
68561 


72877 
72857 
72837 
72817 
72797 


68582172777 
68603172757 
68624  72737 
68645  ;72717 
68666  72697 
68688:72677 
68709'72657 
6873072637 
6875172617 
68772172597 
68793 172677 
6881472657 


68835 
68857 
68878 
68899 


72537 
72617 
72497 
72477 


68920 172467 
68941172437 
0896272417 
68983172397 
69004:72377 
69025|72357 
69040:72337 
69067172317 
6908872297 
69109|72277 
6913072267 
69151 72236 
69172J72216 
6919372196 
6921472176 
69235172156 
6925672136 
69277172116 
69298  72095 
6931972076 
6934072066 
6936172035 
69382;72015 
6940371995 
6942471974 
69445:71954 
69466171934 


Tarn 


N.  cos.lN.sine 


60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 


TABLE  n. 


Log.  Sines  and  Tangents.    (44°)    Natural  Sines. 


65 


0 
1 

2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
26 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


Sine. 


841771 
841902 
842033 
842163 
842294 
842424 
842555 
842686 
842815 
842946 
843076 
843206 
843336 
843466 
843696 
843726 
843855 
843984 
844114 
844243 
844372 
9.844502 
844631 
844760 


845018 
846147 
846276 
845405 
845533 
845662 
845790 
846919 
846047 
846176 
846304 
846432 
846660 
846688 
846816 
846944 
847071 
847199 
847327 
847454 
847582 
847709 
847836 
847964 
848091 
848218 
9.848346 
848472 
848599 
848726 
848862 
848979 
849106 
849232 
849369 
849486 


D.  10" 


21.8 
21.8 
21.8 
21.7 
21.7 
21.7 
21.7 
21.7 
21.7 
21.7 
21.7 
21.6 
21.6 
21.6 
21.6 
21.6 
21.6 
21.6 
21.6 
21.5 
21.5 
21.5 
21.5 
21.6 
21.5 
21.5 
21.5 
21.4 
21.4 
21.4 
21.4 
21.4 
21.4 
21.4 
21.4 
21.4 
21.3 
21.3 
21.3 
21.3 
21.3 
21.3 
21.3 
21.3 
21.2 
21.2 
21.2 
21.2 
21.2 
21.2 
21.2 
21.2 
21.1 
21.1 
21.1 
21.1 
21.1 
21.1 
21.1 
21.1 


Cosine. 


.856934 
866812 
866690 
866668 
866446 
866323 
866201 
866078 
866966 
855833 
856711 
.855588 
855465 
856342 
866219 
865096 
864973 
854850 
854727 
854603 
854480 
.864356 
864233 
864109 
863986 
863862 
863738 
853614 
853490 
863366 
863242 
.863118 
852994 
862869 
852745 
862620 
862496 
862371 
862247 
852122 
861997 
,861872 
861747 
851622 
851497 
861372 
851246 
861121 
860996 
860870 
850746 
860619 
850493 
850368 
850242 
850116 
849990 
849864 
849738 
849611 
849486 


D.  10" 


20 

20 

20 

20 

20 

20 

20. 

20.4 

20.4 

20.4 

20.5 

20.5 

20.5 

20.6 

20.5 

20.5 

20.5 

20.5 

20.6 

20.6 

20.6 

20.6 

20.6 

20.6 

20.6 

20.6 

20.6 

20.7 

20.7 

20.7 

20.7 

20.7 

20.7 

20.7 

20.7 

20.7 

20.8 

20.8 

20.8 

20.8 

20.8 

20.8 

20.8 

20.8 

20.9 

20.9 

20.9 

20.9 

20.9 

20.9 

20.9 

20.9 

21.0 

21.0 

21.0 

21.0 

21.0 

21.0 

21.0 

21.0 


Tang. 


9.984837 
985090 
986343 
986696 
986848 
986101 
986354 
986607 
986860 
987112 
987365 
987618 
987871 
988123 
988376 
988629 
988882 
989134 
989387 
989640 
989893 

9.990146 
990398 
990651 
990903 
991156 
991409 
991662 
991914 
992167 
992420 

9.992672 
992926 
993178 
993430 
993683 
993936 
994189 
994441 
994694 
994947 

9.996199 
996452 
996705 
995957 
996210 
996463 
996715 
996968 
997221 
997473 

9.997726 
997979 
998231 
998484 
998737 
998989 
999242 
999496 
999748 

10.000000 


D.,10' 


42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42 

42 

42 

42 

42 

42 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 


42 

42 

42 

42 

42 

42 

42 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 


Co  tang. 


10.015163 
014910 
014657 
014404 
014162 
013899 
013646 
013393 
013140 
012888 
012635 

iO. 012382 
012129 
011877 
011624 
011371 
011118 
010866 
010613 
010360 
010107 

10.009856 
009602 
009349 
009097 
008844 
008591 
008338 
008086 
007833 
007580 

10- 007328 
007075 
006822 
006670 
006317 
006064 
006811 
005559 
005306 
006063 

10-004801 
004548 
004295 
004043 
003790 
003537  I 
00328^ 
003032 
002779 
002627 

10.002274 
002021 
001769 
001516 
001263 
001011 
000758 
000505 
000263 
000000 


N.  sine.  N.  cos 


69466 
69487 
69608 
69629 
69549 
69670 
69691 
69612 
69633 
69664 
69675 
69696 
69717 
69737 
69758 
69779 
69800 
69821 
69842 
69862 
1 69883 
69904 
69925 
69946 
69966 
69987 
70008 
70029 
70049 
70070 
70091 
70112 
70132 
70153 
70174 
70196 
70215 
70236 
70257 
70277 
70298 
70319 
70339 
70360 
70381 
70401 
70422 
I  70443 
70463 
70484 
70505 
70525 
70546 
70667 
70587 
70608 
70628 
70649 
70670 
70690 
70711 


71934 
71914 
71894 
71873 
71853 
71833 
71813 
71792 
71772 
71752 
71732 
71711 
71691 
71671 
71650 
71630 
71610 
71590 
71569 
71549 
71629 
71608 
71488 
71468 
71447 
71427 
71407 
71386 
71366 
71346 
71325 
71305 
71284 
71264 
71243 
71223 
71203 
71182 
71162 
71141 
71121 
71100 
71080 
71059 
71039 
71019 

0998 
70978 
70967 
70937 

0916 
70896 
70876 

0865 
70834 
70813 
70793 

0772 
70762 
70731 
70711 


Cosine. 


Sine. 


Cotang. 


Tang. 


N.  cos.  N.pine. 


4&  Degrees. 


f  OF  THE  X^ 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  50  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $1.00  ON  THE  SEVENTH  DAY 
OVERDUE. 


NOV  26 1946 


NOV  27 194fi 


t^w* 


^^A 


mm'^m 


SEC'D  LP 


unr~4 1357 


^p^^| 


C^'LiLD 


WIW 


%\\%^t 


LD  21-100m-12,'43  (8796s) 


.11111.11  niiliupiu^ji  I  iiJiii  i^iuiuiiipiimiiilili 


A 


111892 


•■> 


